Articles

15.5E: Exercises for Section 15.5


In exercises 1 - 8, evaluate the triple integrals (displaystyle iiint_E f(x,y,z) , dV) over the solid (E).

1. (f(x,y,z) = z, quad B = ig{(x,y,z), | ,x^2 + y^2 leq 9, quad x leq 0, quad y leq 0, quad 0 leq z leq 1ig})

Answer:
(frac{9pi}{8})

2. (f(x,y,z) = xz^2, space B = ig{(x,y,z), | ,x^2 + y^2 leq 16, space x geq 0, space y leq 0, space -1 leq z leq 1ig})

3. (f(x,y,z) = xy, space B = ig{(x,y,z), | ,x^2 + y^2 leq 1, space x geq 0, space x geq y, space -1 leq z leq 1ig})

Answer:
(frac{1}{8})

4. (f(x,y,z) = x^2 + y^2, space B = ig{(x,y,z), | ,x^2 + y^2 leq 4, space x geq 0, space x leq y, space 0 leq z leq 3ig})

5. (f(x,y,z) = e^{sqrt{x^2+y^2}}, space B = ig{(x,y,z), | ,1 leq x^2 + y^2 leq 4, space y leq 0, space x leq ysqrt{3}, space 2 leq z leq 3 ig})

Answer:
(frac{pi e^2}{6})

6. (f(x,y,z) = sqrt{x^2 + y^2}, space B = ig{(x,y,z), | ,1 leq x^2 + y^2 leq 9, space y leq 0, space 0 leq z leq 1ig})

7. a. Let (B) be a cylindrical shell with inner radius (a) outer radius (b), and height (c) where (0 < a < b) and (c>0). Assume that a function (F) defined on (B) can be expressed in cylindrical coordinates as (F(x,y,z) = f(r) + h(z)), where (f) and (h) are differentiable functions. If (displaystyle int_a^b ar{f} (r) ,dr = 0) and (ar{h}(0) = 0), where (ar{f}) and (ar{h}) are antiderivatives of (f) and (h), respectively, show that (displaystyle iiint_B F(x,y,z) ,dV = 2pi c (bar{f} (b) - a ar{f}(a)) + pi(b^2 - a^2) ar{h} (c).)

b. Use the previous result to show that ( displaystyle iiint_B left(z + sin sqrt{x^2 + y^2} ight) ,dx space dy space dz = 6 pi^2 ( pi - 2),) where (B) is a cylindrical shell with inner radius (pi) outer radius (2pi), and height (2).

8. Let (B) be a cylindrical shell with inner radius (a) outer radius (b) and height (c) where (0 < a < b) and (c > 0). Assume that a function (F) defined on (B) can be expressed in cylindrical coordinates as (F(x,y,z) = f(r) g( heta) f(z)), where (f, space g,) and (h) are differentiable functions. If (displaystyleint_a^b ilde{f} (r) , dr = 0,) where ( ilde{f}) is an antiderivative of (f), show that (displaystyleiiint_B F (x,y,z),dV = [b ilde{f}(b) - a ilde{f}(a)] [ ilde{g}(2pi) - ilde{g}(0)] [ ilde{h}(c) - ilde{h}(0)],) where ( ilde{g}) and ( ilde{h}) are antiderivatives of (g) and (h), respectively.

b. Use the previous result to show that (displaystyleiiint_B z sin sqrt{x^2 + y^2} ,dx space dy space dz = - 12 pi^2,) where (B) is a cylindrical shell with inner radius (pi) outer radius (2pi), and height (2).

In exercises 9 - 12, the boundaries of the solid (E) are given in cylindrical coordinates.

a. Express the region (E) in cylindrical coordinates.

b. Convert the integral (displaystyle iiint_E f(x,y,z) ,dV) to cylindrical coordinates.

9. (E) is bounded by the right circular cylinder (r = 4 sin heta), the (r heta)-plane, and the sphere (r^2 + z^2 = 16).

Answer:

a. (E = ig{(r, heta,z), | ,0 leq heta leq pi, space 0 leq r leq 4 sin heta, space 0 leq z leq sqrt{16 - r^2}ig})

b. (displaystyleint_0^{pi} int_0^{4 sin heta} int_0^{sqrt{16-r^2}} f(r, heta, z) r , dz space dr space d heta)

10. (E) is bounded by the right circular cylinder (r = cos heta), the (r heta)-plane, and the sphere (r^2 + z^2 = 9).

11. (E) is located in the first octant and is bounded by the circular paraboloid (z = 9 - 3r^2), the cylinder (r = sqrt{3}), and the plane (r(cos heta + sin heta) = 20 - z).

Answer:

a. (E = ig{(r, heta,z) , | , 0 leq heta leq frac{pi}{2}, space 0 leq r leq sqrt{3}, space 9 - r^2 leq z leq 10 - r(cos heta + sin heta)ig})

b. (displaystyleint_0^{pi/2} int_0^{sqrt{3}} int_{9-r^2}^{10-r(cos heta + sin heta)} f(r, heta,z) r space dz space dr space d heta)

12. (E) is located in the first octant outside the circular paraboloid (z = 10 - 2r^2) and inside the cylinder (r = sqrt{5}) and is bounded also by the planes (z = 20) and ( heta = frac{pi}{4}).

In exercises 13 - 16, the function (f) and region (E) are given.

a. Express the region (E) and the function (f) in cylindrical coordinates.

b. Convert the integral (displaystyle iiint_B f(x,y,z) ,dV) into cylindrical coordinates and evaluate it.

13. (f(x,y,z) = x^2 + y^2), (E = ig{(x,y,z), | ,0 leq x^2 + y^2 leq 9, space x geq 0, space y geq 0, space 0 leq z leq x + 3ig})

Answer:

a. (E = ig{(r, heta,z), | ,0 leq r leq 3, space 0 leq heta leq frac{pi}{2}, space 0 leq z leq r space cos heta + 3ig},)
(f(r, heta,z) = frac{1}{r space cos heta + 3})

b. (displaystyle int_0^3 int_0^{pi/2} int_0^{r space cos heta+3} frac{r}{r space cos heta + 3} , dz space d heta space dr = frac{9pi}{4})

14. (f(x,y,z) = x^2 + y^2, space E = ig{(x,y,z) |0 leq x^2 + y^2 leq 4, space y geq 0, space 0 leq z leq 3 - x ig})

15. (f(x,y,z) = x, space E = ig{(x,y,z), | ,1 leq y^2 + z^2 leq 9, space 0 leq x leq 1 - y^2 - z^2ig})

Answer:

a. (y = r space cos heta, space z = r space sin heta, space x = z,space E = ig{(r, heta,z), | ,1 leq r leq 3, space 0 leq heta leq 2pi, space 0 leq z leq 1 - r^2ig}, space f(r, heta,z) = z);

b. (displaystyle int_1^3 int_0^{2pi} int_0^{1-r^2} z r space dz space d heta space dr = frac{356 pi}{3})

16. (f(x,y,z) = y, space E = ig{(x,y,z), | ,1 leq x^2 + z^2 leq 9, space 0 leq y leq 1 - x^2 - z^2 ig})

In exercises 17 - 24, find the volume of the solid (E) whose boundaries are given in rectangular coordinates.

17. (E) is above the (xy)-plane, inside the cylinder (x^2 + y^2 = 1), and below the plane (z = 1).

Answer:
(pi)

18. (E) is below the plane (z = 1) and inside the paraboloid (z = x^2 + y^2).

19. (E) is bounded by the circular cone (z = sqrt{x^2 + y^2}) and (z = 1).

Answer:
(frac{pi}{3})

20. (E) is located above the (xy)-plane, below (z = 1), outside the one-sheeted hyperboloid (x^2 + y^2 - z^2 = 1), and inside the cylinder (x^2 + y^2 = 2).

21. (E) is located inside the cylinder (x^2 + y^2 = 1) and between the circular paraboloids (z = 1 - x^2 - y^2) and (z = x^2 + y^2).

Answer:
(pi)

22. (E) is located inside the sphere (x^2 + y^2 + z^2 = 1), above the (xy)-plane, and inside the circular cone (z = sqrt{x^2 + y^2}).

23. (E) is located outside the circular cone (x^2 + y^2 = (z - 1)^2) and between the planes (z = 0) and (z = 2).

Answer:
(frac{4pi}{3})

24. (E) is located outside the circular cone (z = 1 - sqrt{x^2 + y^2}), above the (xy)-plane, below the circular paraboloid, and between the planes (z = 0) and (z = 2).

25. [T] Use a computer algebra system (CAS) to graph the solid whose volume is given by the iterated integral in cylindrical coordinates (displaystyle int_{-pi/2}^{pi/2} int_0^1 int_{r^2}^r r , dz , dr , d heta.) Find the volume (V) of the solid. Round your answer to four decimal places.

Answer:

(V = frac{pi}{12} approx 0.2618)

26. [T] Use a CAS to graph the solid whose volume is given by the iterated integral in cylindrical coordinates (displaystyle int_0^{pi/2} int_0^1 int_{r^4}^r r , dz , dr , d heta.) Find the volume (E) of the solid. Round your answer to four decimal places.

27. Convert the integral (displaystyleint_0^1 int_{-sqrt{1-y^2}}^{sqrt{1-y^2}} int_{x^2+y^2}^{sqrt{x^2+y^2}} xz space dz space dx space dy) into an integral in cylindrical coordinates.

Answer:
(displaystyleint_0^1 int_0^{pi} int_{r^2}^r zr^2 space cos heta , dz space d heta space dr)

28. Convert the integral (displaystyle int_0^2 int_0^y int_0^1 (xy + z) , dz space dx space dy) into an integral in cylindrical coordinates.

In exercises 29 - 32, evaluate the triple integral (displaystyle iiint_B f(x,y,z) ,dV) over the solid (B).

29. (f(x,y,z) = 1, space B = ig{(x,y,z), | ,x^2 + y^2 + z^2 leq 90, space z geq 0ig})

[Hide Solution]

Answer:
(180 pi sqrt{10})

30. (f(x,y,z) = 1 - sqrt{x^2 + y^2 + z^2}, space B = ig{(x,y,z), | ,x^2 + y^2 + z^2 leq 9, space y geq 0, space z geq 0ig})

31. (f(x,y,z) = sqrt{x^2 + y^2}, space B ) is bounded above by the half-sphere (x^2 + y^2 + z^2 = 9) with (z geq 0) and below by the cone (2z^2 = x^2 + y^2).

Answer:
(frac{81pi(pi - 2)}{16})

32. (f(x,y,z) = sqrt{x^2 + y^2}, space B ) is bounded above by the half-sphere (x^2 + y^2 + z^2 = 16) with (z geq 0) and below by the cone (2z^2 = x^2 + y^2).

33. Show that if (F ( ho, heta,varphi) = f( ho)g( heta)h(varphi)) is a continuous function on the spherical box (B = ig{( ho, heta,varphi), | ,a leq ho leq b, space alpha leq heta leq eta, space gamma leq varphi leq psiig}), then (displaystyleiiint_B F space dV = left(int_a^b ho^2 f( ho) space dr ight) left( int_{alpha}^{eta} g ( heta) space d heta ight)left( int_{gamma}^{psi} h (varphi) space sin varphi space dvarphi ight).)

34. A function (F) is said to have spherical symmetry if it depends on the distance to the origin only, that is, it can be expressed in spherical coordinates as (F(x,y,z) = f( ho)), where ( ho = sqrt{x^2 + y^2 + z^2}). Show that (displaystyleiiint_B F(x,y,z) ,dV = 2pi int_a^b ho^2 f( ho) ,d ho,) where (B) is the region between the upper concentric hemispheres of radii (a) and (b) centered at the origin, with (0 < a < b) and (F) a spherical function defined on (B).

Use the previous result to show that (displaystyleiiint_B (x^2 + y^2 + z^2) sqrt{x^2 + y^2 + z^2} dV = 21 pi,) where (B = ig{(x,y,z), | ,1 leq x^2 + y^2 + z^2 leq 2, space z geq 0ig}).

35. Let (B) be the region between the upper concentric hemispheres of radii a and b centered at the origin and situated in the first octant, where (0 < a < b). Consider F a function defined on B whose form in spherical coordinates (( ho, heta,varphi)) is (F(x,y,z) = f( ho)cos varphi). Show that if (g(a) = g(b) = 0) and (displaystyleint_a^b h ( ho) , d ho = 0,) then (displaystyleiiint_B F(x,y,z),dV = frac{pi^2}{4} [ah(a) - bh(b)],) where (g) is an antiderivative of (f) and (h) is an antiderivative of (g).

Use the previous result to show that (displaystyle iiint_B = frac{z cos sqrt{x^2 + y^2 + z^2}}{sqrt{x^2 + y^2 + z^2}} , dV = frac{3pi^2}{2},) where (B) is the region between the upper concentric hemispheres of radii (pi) and (2pi) centered at the origin and situated in the first octant.

In exercises 36 - 39, the function (f) and region (E) are given.

a. Express the region (E) and function (f) in cylindrical coordinates.

b. Convert the integral (displaystyle iiint_B f(x,y,z), dV) into cylindrical coordinates and evaluate it.

36. (f(x,y,z) = z; space E = ig{(x,y,z), | ,0 leq x^2 + y^2 + z^2 leq 1, space z geq 0ig})

37. (f(x,y,z) = x + y; space E = ig{(x,y,z), | ,1 leq x^2 + y^2 + z^2 leq 2, space z geq 0, space y geq 0ig})

Answer:

a. (f( ho, heta, varphi) = ho space sin varphi space (cos heta + sin heta), space E = ig{( ho, heta,varphi), | ,1 leq ho leq 2, space 0 leq heta leq pi, space 0 leq varphi leq frac{pi}{2}ig});

b. (displaystyle int_0^{pi} int_0^{pi/2} int_1^2 ho^3 cos varphi space sin varphi space d ho space dvarphi space d heta = frac{15pi}{8})

38. (f(x,y,z) = 2xy; space E = ig{(x,y,z), | ,sqrt{x^2 + y^2} leq z leq sqrt{1 - x^2 - y^2}, space x geq 0, space y geq 0ig})

39. (f(x,y,z) = z; space E = ig{(x,y,z), | ,x^2 + y^2 + z^2 - 2x leq 0, space sqrt{x^2 + y^2} leq zig})

Answer:

a. (f( ho, heta,varphi) = ho space cos varphi; space E = ig{( ho, heta,varphi), | ,0 leq ho leq 2 space cos varphi, space 0 leq heta leq frac{pi}{2}, space 0 leq varphi leq frac{pi}{4}ig});

b. (displaystyleint_0^{pi/2} int_0^{pi/4} int_0^{2 space cos varphi} ho^3 sin varphi space cos varphi space d ho space dvarphi space d heta = frac{7pi}{24})

In exercises 40 - 41, find the volume of the solid (E) whose boundaries are given in rectangular coordinates.

40. (E = ig{ (x,y,z), | ,sqrt{x^2 + y^2} leq z leq sqrt{16 - x^2 - y^2}, space x geq 0, space y geq 0ig})

41. (E = ig{ (x,y,z), | ,x^2 + y^2 + z^2 - 2z leq 0, space sqrt{x^2 + y^2} leq zig})

Answer:
(frac{pi}{4})

42. Use spherical coordinates to find the volume of the solid situated outside the sphere ( ho = 1) and inside the sphere ( ho = cos varphi), with (varphi in [0,frac{pi}{2}]).

43. Use spherical coordinates to find the volume of the ball ( ho leq 3) that is situated between the cones (varphi = frac{pi}{4}) and (varphi = frac{pi}{3}).

Answer:
(9pi (sqrt{2} - 1))

44. Convert the integral (displaystyle int_{-4}^4 int_{-sqrt{16-y^2}}^{sqrt{16-y^2}} int_{-sqrt{16-x^2-y^2}}^{sqrt{16-x^2-y^2}} (x^2 + y^2 + z^2) , dz , dx , dy) into an integral in spherical coordinates.

45. Convert the integral (displaystyle int_0^4 int_0^{sqrt{16-x^2}} int_{-sqrt{16-x^2-y^2}}^{sqrt{16-x^2-y^2}} (x^2 + y^2 + z^2)^2 , dz space dy space dx) into an integral in spherical coordinates.

Answer:
(displaystyleint_0^{pi/2} int_0^{pi/2} int_0^4 ho^6 sin varphi , d ho , dphi , d heta)

47. [T] Use a CAS to graph the solid whose volume is given by the iterated integral in spherical coordinates (displaystyle int_{pi/2}^{pi} int_{5pi}^{pi/6} int_0^2 ho^2 sin varphi space d ho space dvarphi space d heta.) Find the volume (V) of the solid. Round your answer to three decimal places.

Answer:

(V = frac{4pisqrt{3}}{3} approx 7.255)

48. [T] Use a CAS to graph the solid whose volume is given by the iterated integral in spherical coordinates as (displaystyle int_0^{2pi} int_{3pi/4}^{pi/4} int_0^1 ho^2 sin varphi space d ho space dvarphi space d heta.) Find the volume (V) of the solid. Round your answer to three decimal places.

49. [T] Use a CAS to evaluate the integral (displaystyle iiint_E (x^2 + y^2) , dV) where (E) lies above the paraboloid (z = x^2 + y^2) and below the plane (z = 3y).

Answer:
(frac{343pi}{32})

50. [T]

a. Evaluate the integral (displaystyle iiint_E e^{sqrt{x^2+y^2+z^2}}, dV,) where (E) is bounded by spheres (4x^2 + 4y^2 + 4z^2 = 1) and (x^2 + y^2 + z^2 = 1).

b. Use a CAS to find an approximation of the previous integral. Round your answer to two decimal places.

51. Express the volume of the solid inside the sphere (x^2 + y^2 + z^2 = 16) and outside the cylinder (x^2 + y^2 = 4) as triple integrals in cylindrical coordinates and spherical coordinates, respectively.

Answer:
(displaystyle int_0^{2pi}int_2^4int_{−sqrt{16−r^2}}^{sqrt{16−r^2}}r,dz,dr,dθ) and (displaystyle int_{pi/6}^{5pi/6}int_0^{2pi}int_{2csc phi}^{4} ho^2sin ho , d ho , d heta , dphi)

52. Express the volume of the solid inside the sphere (x^2 + y^2 + z^2 = 16) and outside the cylinder (x^2 + y^2 = 4) that is located in the first octant as triple integrals in cylindrical coordinates and spherical coordinates, respectively.

53. The power emitted by an antenna has a power density per unit volume given in spherical coordinates by (p( ho, heta,varphi) = frac{P_0}{ ho^2} cos^2 heta space sin^4 varphi), where (P_0) is a constant with units in watts. The total power within a sphere (B) of radius (r) meters is defined as (displaystyle P = iiint_B p( ho, heta,varphi) , dV.) Find the total power (P).

Answer:
(P = frac{32P_0 pi}{3}) watts

54. Use the preceding exercise to find the total power within a sphere (B) of radius 5 meters when the power density per unit volume is given by (p( ho, heta,varphi) = frac{30}{ ho^2} cos^2 heta sin^4 varphi).

55. A charge cloud contained in a sphere (B) of radius (r) centimeters centered at the origin has its charge density given by (q(x,y,z) = ksqrt{x^2 + y^2 + z^2}frac{mu C}{cm^3}), where (k > 0). The total charge contained in (B) is given by (displaystyle Q = iiint_B q(x,y,z) , dV.) Find the total charge (Q).

Answer:
(Q = kr^4 pi mu C)

56. Use the preceding exercise to find the total charge cloud contained in the unit sphere if the charge density is (q(x,y,z) = 20 sqrt{x^2 + y^2 + z^2} frac{mu C}{cm^3}).

Contributors

Gilbert Strang (MIT) and Edwin “Jed” Herman (Harvey Mudd) with many contributing authors. This content by OpenStax is licensed with a CC-BY-SA-NC 4.0 license. Download for free at http://cnx.org.


West Virginia Code Chapter 15. Public Safety § 15-5-6. Emergency powers of Governor

(a) The provisions of this section are operative only during the existence of a state of emergency or state of preparedness. The existence of a state of emergency or state of preparedness may be proclaimed by the Governor or by concurrent resolution of the Legislature if the Governor in the proclamation, or the Legislature in the resolution, finds that an attack upon the United States has occurred or is anticipated in the immediate future, or that a natural or man-made disaster of major proportions has actually occurred or is imminent within the state, or that an emergency exists or may be imminent due to a large-scale threat beyond local control, and that the safety and welfare of the inhabitants of this state require an invocation of the provisions of this section.

(b) Any state of emergency or state of preparedness, whether proclaimed by the Governor or by the Legislature, terminates upon the proclamation of the termination by the Governor, or the passage by the Legislature of a concurrent resolution terminating the state of emergency or state of preparedness: Provided, That in no case shall a state of preparedness last longer than thirty days.

(c) So long as a state of emergency or state of preparedness exists, the Governor has and may exercise the following additional emergency powers:

(1) To enforce all laws and rules relating to the provision of emergency services and to assume direct operational control of any or all emergency service forces and helpers in the state

(2) To sell, lend, lease, give, transfer or deliver materials or perform functions relating to emergency services on terms and conditions he or she prescribes and without regard to the limitations of any existing law and to account to the State Treasurer for any funds received for the property

(3) To procure materials and facilities for emergency services by purchase, condemnation under the provisions of chapter fifty-four of this code or seizure pending institution of condemnation proceedings within thirty days from the seizing thereof and to construct, lease, transport, store, maintain, renovate or distribute the materials and facilities. Compensation for property so procured shall be made in the manner provided in chapter fifty-four of this code

(4) To obtain the services of necessary personnel, required during the emergency, and to compensate them for their services from his or her contingent funds or other funds available to him or her

(5) To provide and compel the evacuation of all or part of the population from any stricken or threatened area within the state and to take steps that are necessary for the receipt and care of the evacuees

(6) To control ingress and egress to and from a disaster area or an area where large-scale threat exists, the movement of persons within the area and the occupancy of premises therein

(7) To suspend the provisions of any regulatory statute prescribing the procedures for conduct of state business or the orders, rules of any state agency, if strict compliance therewith would in any way prevent, hinder or delay necessary action in coping with the emergency

(8) To use available resources of the state and of its political subdivisions that are reasonably necessary to cope with the emergency

(9) To suspend or limit the sale, dispensing or transportation of alcoholic beverages, explosives and combustibles

(10) To make provision for the availability and use of temporary emergency housing and

(11) To perform and exercise other functions, powers and duties that are necessary to promote and secure the safety and protection of the civilian population.

(d) The declaration of a state of preparedness has the same effect as a declaration of a state of emergency for the purposes of the Emergency Management Assistance Compact established in section twenty-two of this article and the Statewide Mutual Aid Systems set forth in section twenty-eight of this article.

(e) The powers granted under this section do not authorize any action that would violate the prohibitions of section nineteen-a of this article.


5.3 Testimony by Employees and Production of Documents Where the United States is Not a Party

Information about requesting the testimony of an ORR employee or production of documents where the United States is not a party is available at 45 CFR Part 2 Visit disclaimer page .

5.3.1 Care Provider Testimony and Views

ORR care provider staff should follow their organization’s policy regarding testimony. This may include talking to the care provider’s attorney. If care provider staff do testify, they must make clear that they do not represent ORR and that their testimony does not reflect the opinion or position of ORR.

5.3.2 Confidentiality of Information

Care providers must maintain the confidentiality of children’s information, and must protect it from unauthorized disclosure. Therefore, as a general matter, they may not reveal client information in affidavits or testimony, where not legally required to so, and without ORR approval. However, care providers may allow a youth 14 and older, who wishes the staff to testify or provide a declaration, to consent, in writing, to the testimony, and then ORR could approve the disclosure of the confidential information.

For children younger than 14, or who do not have the capacity to consent, the care provider should seek the written consent of the child’s parent or legal guardian in order to include confidential information in an affidavit or written or verbal testimony. If such consent is not available, the care provider should contact ORR.

5.3.3 Release of Records Without Prior Approval

The records of unaccompanied alien children are the property of ORR, and care providers may not release records without prior approval from ORR.


15.5E: Exercises for Section 15.5

Janet knew that her argument was really weak. She kept looking at the data trying to find a way around the weakness. Finally, it hit her. She realized that she could hide the weakest part of her argument in a really complex presentation aid. If the people can’t understand it, they can’t use it against me, she thought to herself.

While she was nervous during her presentation, she was confident that no one would notice what she did. Thankfully, at the end of her presentation everyone applauded. During the question and answer period that followed, no one questioned the weak information. In fact, no one seemed to even remember the presentation aid at all.

  1. Is hiding weak information in a complex presentation aid ethical?
  2. Are complex aids that don’t lead to audience understanding ever ethical?
  3. If you were Janet’s boss and you found out what she had done, would you think she was an unethical person or just a good, albeit manipulative, speaker?

End-of-Chapter Assessment

Polly was in the middle of her speech about the importance of climate change. The presentation aid she shows is a picture outlining where the hole in the earth’s ozone is located. What aspect of audience understanding is Polly hoping to impact with her aid?

Benny conducted a simple survey of his fraternity members to see what their thoughts were on instituting a no-hazing policy. During his presentation to the group he used the following aid to discuss his findings. What type of aid is Benny using?

Which form of presentation aids are drawings that outline and explain the parts of an object, process, or phenomenon that cannot be readily seen.

Which of the following is not true about using black/dry-erase boards as presentation aids?

  1. Don’t write in complete sentences.
  2. Never write in cursive.
  3. Write large enough so everyone in the room can see.
  4. Make sure your handwriting is legible.
  5. Black/dry-erase boards are appropriate for every speech context.

Which of the following is a tip for effectively using presentation aids?


New York Consolidated Laws, General Obligations Law - GOB § 5-1514. Certain gift transactions formal requirements statutory form

1. If the principal intends to authorize the agent to make gifts other than gifts authorized by subdivision fourteen of section 5-1502I of this title, the principal must expressly grant such authority either in a statutory gifts rider to a statutory short form power of attorney or in a non-statutory power of attorney executed pursuant to the requirements of paragraph (b) of subdivision nine of this section.

2. The principal may authorize the agent to make gifts to the principal's spouse, children and more remote descendents, and parents, not to exceed, for each donee, the annual federal gift tax exclusion amount pursuant to the Internal Revenue Code. 1 For gifts to the principal's children and more remote descendants, and parents, the maximum amount of the gift to each donee shall not exceed twice the gift tax exclusion amount, if the principal's spouse agrees to split gift treatment pursuant to the Internal Revenue Code.

3. The principal may also authorize the agent to:

(a) make gifts up to a specified dollar amount, or unlimited in amount

(b) make gifts to any person or persons

(c) make gifts in any of the following ways:

(1) opening, modifying or terminating a deposit account in the name of the principal and other joint tenants

(2) opening, modifying or terminating any other joint account in the name of the principal and other joint tenants

(3) opening, modifying or terminating a bank account in trust form as described in section 7-5.1 of the estates, powers and trusts law , and designate or change the beneficiary or beneficiaries of such account

(4) opening, modifying or terminating a transfer on death account as described in part four of article thirteen of the estates, powers and trusts law, and designate or change the beneficiary or beneficiaries of such account

(5) changing the beneficiary or beneficiaries of any contract of insurance on the life of the principal or annuity contract for the benefit of the principal

(6) procuring new, different or additional contracts of insurance on the life of the principal or annuity contracts for the benefit of the principal and designate the beneficiary or beneficiaries of any such contract

(7) designating or changing the beneficiary or beneficiaries of any type of retirement benefit or plan

(8) creating, amending, revoking or terminating an inter vivos trust and

(9) opening, modifying or terminating other property interests or rights of survivorship, and designate or change the beneficiary or beneficiaries therein.

A gift to an individual authorized by this subdivision may be made outright, by exercise or release of a presently exercisable general or special power of appointment held by the principal, to a trust established or created for such individual, to a Uniform Transfers to Minors Act account for such individual (regardless of who is the custodian), or to a tuition savings account or prepaid tuition plan as defined under section 529 of the Internal Revenue Code for the benefit of such individual (without regard to who is the account owner or responsible individual for such account).

(a) exercise any authority described in subdivision two or three of this section unless such authority is expressly granted in a statutory gifts rider to a statutory short form power of attorney or in a non-statutory power of attorney executed pursuant to the requirements of paragraph (b) of subdivision nine of this section

(b) make a gift to himself or herself or create in himself or herself an interest in the principal's property pursuant to any grant of authority described in subdivision two or three of this section unless such authority is expressly granted in a statutory gifts rider to a statutory short form power of attorney or in a non-statutory power of attorney executed pursuant to the requirements of paragraph (b) of subdivision nine of this section.

5. Any authority granted to an agent pursuant to subdivision two or three or paragraph (b) of subdivision four of this section must be exercised according to any instructions in this document or in any other writing provided by the principal regarding the exercise of any authority, or otherwise for purposes which the agent reasonably deems to be in the best interest of the principal, specifically including financial, estate, or tax planning, including minimization of income, estate, inheritance, generation-skipping transfer or gift taxes.

6. Construction of the provisions of the statutory gifts rider. (a) In a statutory gifts rider to a statutory short form power of attorney, the language “I grant authority to my agent to make gifts to my spouse, children and more remote descendants, and parents, not to exceed, for each donee, the annual federal gift tax exclusion amount pursuant to the Internal Revenue Code. For gifts to my children and more remote descendants, and parents, the maximum amount of the gift to each donee shall not exceed twice the gift tax exclusion amount, if my spouse agrees to split gift treatment pursuant to the Internal Revenue Code” must be construed to mean that the principal authorizes the agent:

(1) To make gifts on behalf of the principal to the principal's spouse, children and other descendants, and parents. Gifts to a donee shall not exceed in any calendar year the amount of the federal gift tax exclusion available to the principal under section 2503(b) of the Internal Revenue Code . Gifts may be made outright or by exercise or release of a presently exercisable general power or special power of appointment held by the principal, to a trust established or created for such individual (provided that gifts to such trust qualify for the federal gift tax exclusion under section 2503(b) or (c) of the Internal Revenue Code ), to a Uniform Transfers to Minors Act account for such individual (regardless of who is the custodian), to a tuition savings account or prepaid tuition plan as defined under section 529 of the Internal Revenue Code for the benefit of such individual (without regard to who is the account owner of or responsible person for such account)

(2) To make gifts up to twice the annual federal gift tax exclusion amount on behalf of both the principal and the principal's spouse, to the principal's children and other descendants, and parents, if the principal's spouse consents to the splitting of such gifts pursuant to section 2513 of the Internal Revenue Code

(3) To consent, pursuant to Section 2513(a) of the Internal Revenue Code , to the splitting of gifts made by the principal's spouse to the principal's children and other descendants in any amount, and to the splitting of gifts made by the principal's spouse to any other persons in amounts not exceeding the aggregate annual gift tax exclusions for both spouses under Section 2503(b) of said Code (or cognate provisions of any successor statute) and

(4) To satisfy pledges made to organizations, whether charitable or otherwise, by the principal.

(b) Any authority granted to an agent under a statutory gifts rider to a statutory short form power of attorney must be construed to mean that the principal authorizes the agent:

(1) To prepare, execute, consent to on behalf of the principal, and file any return, report, declaration or other document required by the laws of the United States, or by any state or political subdivision thereof, or by any foreign country or political subdivision thereof, which the agent deems to be desirable or necessary with respect to any gift made under the authority of this section

(2) To execute, acknowledge, seal and deliver any deed, assignment, agreement, trust agreement, authorization, check, or other instrument which the agent deems useful for the accomplishment of any of the purposes enumerated in this section

(3) To prosecute, defend, submit to alternative dispute resolution, settle and propose or accept a compromise with respect to any claim existing in favor of or against the principal based on or involving any gift transaction or to intervene in any related action or proceeding

(4) To hire, discharge and compensate any attorney, accountant, expert witness, or other assistant or assistants when the agent deems that action to be desirable for the proper execution by the agent of any of the authorities described in this section, and for the keeping of needed records thereof and

(5) In general, and in addition to but not in contravention of all the specific acts listed in this section, to do any other act or acts which the agent deems desirable or necessary to complete any such gift on behalf of the principal.

(c) The authority explicitly authorized in this section shall be construed to include any like authority authorized in any other section of this title. Accordingly, such like authorities as are authorized in any other section of this title may not be exercised by the agent unless they are expressly granted to the agent in the statutory gifts rider or in a non-statutory power of attorney executed pursuant to the requirements of paragraph (b) of subdivision nine of this section.

(d) The statutory gifts rider may be modified pursuant to section 5-1503 of this title to contain additional provisions authorizing the agent to make any or all of the transactions specified in subdivision three of this section.

7. All authority described in this section shall be exercisable equally with respect to a gift of any property in which the principal is interested at the time the power of attorney is given or in which the principal becomes interested after that time, and whether located in this state or elsewhere.

8. If, after naming the spouse as a permissible recipient of gifting, the principal's marriage is terminated by divorce or annulment, as defined in subparagraph two of paragraph (f) of section 5-1.4 of the estates, powers and trusts law , the divorce or annulment revokes the authority to gift to the former spouse, unless the statutory gifts rider or the non-statutory power of attorney executed pursuant to the requirements of paragraph (b) of subdivision nine of this section expressly provides otherwise. If the authority to gift to the former spouse is revoked solely by this subdivision, it shall be revived by the principal's remarriage to the former spouse.

9. To be valid, a statutory gifts rider to a statutory short form power of attorney must:

(a) Be typed or printed using letters which are legible or of clear type no less than twelve point in size, or, if in writing, a reasonable equivalent thereof.

(b) Be signed and dated by a principal with capacity, with the signature of the principal duly acknowledged in the manner prescribed for the acknowledgment of a conveyance of real property, and witnessed by two persons who are not named in the instrument as permissible recipients of gifts, in the manner described in subparagraph two of paragraph (a) of section 3-2.1 of the estates, powers and trusts law . The person who takes the acknowledgment, under this paragraph, may also serve as one of the witnesses.

(c) Be accompanied by a statutory short form power of attorney in which the authority (SGR) is initialed by the principal.

(d) Be executed simultaneously with the statutory short form power of attorney and in the manner provided in this section.

10. The use of the following shall be construed as the “Statutory Gifts Rider” for a statutory short form power of attorney:

NEW YORK STATUTORY GIFTS RIDER

FOR CERTAIN GIFT TRANSACTIONS

CAUTION TO THE PRINCIPAL: This OPTIONAL rider allows you to authorize your agent to make gifts in excess of an annual total of $500 for all gifts described in (I) of the Grant of Authority section of the statutory short form Power of Attorney (under personal and family maintenance), or certain other gift transactions during your lifetime. You do not have to execute this rider if you only want your agent to make gifts described in (I) of the Grant of Authority section of the statutory short form Power of Attorney and you initialed “(I)” on that section of that form. Granting any of the following authority to your agent gives your agent the authority to take actions which could significantly reduce your property or change how your property is distributed at your death. “Certain gift transactions” are described in section 5-1514 of the General Obligations Law . This Gifts Rider does not require your agent to exercise granted authority, but when he or she exercises this authority, he or she must act according to any instructions you provide, or otherwise in your best interest.

This Gifts Rider and the Power of Attorney it supplements must be read together as a single instrument.

Before signing this document authorizing your agent to make gifts, you should seek legal advice to ensure that your intentions are clearly and properly expressed.

(a) GRANT OF LIMITED AUTHORITY TO MAKE GIFTS

Granting gifting authority to your agent gives your agent the authority to take actions which could significantly reduce your property.

If you wish to allow your agent to make gifts to himself or herself, you must separately grant that authority in subdivision (c) below.

To grant your agent the gifting authority provided below, initial the bracket to the left of the authority.

( ) I grant authority to my agent to make gifts to my spouse, children and more remote descendants, and parents, not to exceed, for each donee, the annual federal gift tax exclusion amount pursuant to the Internal Revenue Code. For gifts to my children and more remote descendants, and parents, the maximum amount of the gift to each donee shall not exceed twice the gift tax exclusion amount, if my spouse agrees to split gift treatment pursuant to the Internal Revenue Code.

This authority must be exercised pursuant to my instructions, or otherwise for purposes which the agent reasonably deems to be in my best interest.

Use this section if you wish to authorize gifts in amounts smaller than the gift tax exclusion amount, in amounts in excess of the gift tax exclusion amount, gifts to other beneficiaries, or other gift transactions.

Granting such authority to your agent gives your agent the authority to take actions which could significantly reduce your property and/or change how your property is distributed at your death. If you wish to authorize your agent to make gifts to himself or herself, you must separately grant that authority in subdivision (c) below.

( ) I grant the following authority to my agent to make gifts pursuant to my instructions, or otherwise for purposes which the agent reasonably deems to be in my best interest:

(c) GRANT OF SPECIFIC AUTHORITY FOR AN AGENT TO MAKE GIFTS TO HIMSELF OR HERSELF: (OPTIONAL)

If you wish to authorize your agent to make gifts to himself or herself, you must grant that authority in this section, indicating to which agent(s) the authorization is granted, and any limitations and guidelines.

( ) I grant specific authority for the following agent(s) to make the following gifts to himself or herself:

This authority must be exercised pursuant to my instructions, or otherwise for purposes which the agent reasonably deems to be in my best interest.

(d) ACCEPTANCE BY THIRD PARTIES: I agree to indemnify the third party for any claims that may arise against the third party because of reliance on this Statutory Gifts Rider.

(e) SIGNATURE OF PRINCIPAL AND ACKNOWLEDGMENT:

In Witness Whereof I have hereunto signed my name on ․․․․․․․․․․․․․․․․․, 20․․․․.

(f) SIGNATURES OF WITNESSES:

By signing as a witness, I acknowledge that the principal signed the Statutory Gifts Rider in my presence and the presence of the other witness, or that the principal acknowledged to me that the principal's signature was affixed by him or her or at his or her direction. I also acknowledge that the principal has stated that this Statutory Gifts Rider reflects his or her wishes and that he or she has signed it voluntarily. I am not named herein as a permissible recipient of gifts.


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The provisions of this title shall not apply to the following powers of attorney:

1. a power of attorney given primarily for a business or commercial purpose, including without limitation:

(a) a power to the extent it is coupled with an interest in the subject of the power

(b) a power given to or for the benefit of a creditor in connection with a loan or other credit transaction

(c) a power given to facilitate transfer or disposition of one or more specific stocks, bonds or other assets, whether real, personal, tangible or intangible

2. a proxy or other delegation to exercise voting rights or management rights with respect to an entity

3. a power created on a form prescribed by a government or governmental subdivision, agency or instrumentality for a governmental purpose

4. a power authorizing a third party to prepare, execute, deliver, submit and/or file a document or instrument with a government or governmental subdivision, agency or instrumentality or other third party

5. a power authorizing a financial institution or employee of a financial institution to take action relating to an account in which the financial institution holds cash, securities, commodities or other financial assets on behalf of the person giving the power

6. a power given by an individual who is or is seeking to become a director, officer, shareholder, employee, partner, limited partner, member, unit owner or manager of a corporation, partnership, limited liability company, condominium or other legal or commercial entity in his or her capacity as such

7. a power contained in a partnership agreement, limited liability company operating agreement, declaration of trust, declaration of condominium, condominium bylaws, condominium offering plan or other agreement or instrument governing the internal affairs of an entity authorizing a director, officer, shareholder, employee, partner, limited partner, member, unit owner, manager or other person to take lawful action relating to such entity

8. a power given to a condominium managing agent to take action in connection with the use, management and operation of a condominium unit

9. a power given to a licensed real estate broker to take action in connection with a listing of real property, mortgage loan, lease or management agreement

10. a power authorizing acceptance of service of process on behalf of the principal and

11. a power created pursuant to authorization provided by a federal or state statute, other than this title, that specifically contemplates creation of the power, including without limitation a power to make health care decisions or decisions involving the disposition of remains.

Nothing in this section shall be deemed to prohibit use of a statutory short form power of attorney or a nonstatutory power of attorney in connection with any of the transactions described in this section.


7.2 Commutative and Associative Properties

In the next few sections, we will take a look at the properties of real numbers. Many of these properties will describe things you already know, but it will help to give names to the properties and define them formally. This way we’ll be able to refer to them and use them as we solve equations in the next chapter.

Use the Commutative and Associative Properties

Think about adding two numbers, such as 5 5 and 3 . 3 .

The results are the same. 5 + 3 = 3 + 5 5 + 3 = 3 + 5

Notice, the order in which we add does not matter. The same is true when multiplying 5 5 and 3 . 3 .

These examples illustrate the commutative properties of addition and multiplication.

Commutative Properties

Commutative Property of Addition : if a a and b b are real numbers, then

Commutative Property of Multiplication : if a a and b b are real numbers, then

The commutative properties have to do with order. If you change the order of the numbers when adding or multiplying, the result is the same.

Example 7.5

Use the commutative properties to rewrite the following expressions:

Solution

Use the commutative properties to rewrite the following:

Use the commutative properties to rewrite the following:

What about subtraction? Does order matter when we subtract numbers? Does 7 − 3 7 − 3 give the same result as 3 − 7 ? 3 − 7 ?

Since changing the order of the subtraction did not give the same result, we can say that subtraction is not commutative.

Let’s see what happens when we divide two numbers. Is division commutative?

Since changing the order of the division did not give the same result, division is not commutative.

Addition and multiplication are commutative. Subtraction and division are not commutative.

Suppose you were asked to simplify this expression.

How would you do it and what would your answer be?

Both ways give the same result, as shown in Figure 7.3. (Remember that parentheses are grouping symbols that indicate which operations should be done first.)

When adding three numbers, changing the grouping of the numbers does not change the result. This is known as the Associative Property of Addition.

The same principle holds true for multiplication as well. Suppose we want to find the value of the following expression:

Changing the grouping of the numbers gives the same result, as shown in Figure 7.4.

When multiplying three numbers, changing the grouping of the numbers does not change the result. This is known as the Associative Property of Multiplication.

If we multiply three numbers, changing the grouping does not affect the product.

You probably know this, but the terminology may be new to you. These examples illustrate the Associative Properties.

Associative Properties

Associative Property of Addition: if a , b , a , b , and c c are real numbers, then

Associative Property of Multiplication: if a , b , a , b , and c c are real numbers, then

Example 7.6

Use the associative properties to rewrite the following:

Solution

Use the associative properties to rewrite the following:
ⓐ ( 1 + 0.7 ) + 0.3 = __________ ( 1 + 0.7 ) + 0.3 = __________ ⓑ ( −9 · 8 ) · 3 4 = __________ ( −9 · 8 ) · 3 4 = __________

Use the associative properties to rewrite the following:
ⓐ ( 4 + 0.6 ) + 0.4 = __________ ( 4 + 0.6 ) + 0.4 = __________ ⓑ ( −2 · 12 ) · 5 6 = __________ ( −2 · 12 ) · 5 6 = __________

Besides using the associative properties to make calculations easier, we will often use it to simplify expressions with variables.

Example 7.7

Use the Associative Property of Multiplication to simplify: 6 ( 3 x ) . 6 ( 3 x ) .

Solution

Use the Associative Property of Multiplication to simplify the given expression: 8 ( 4 x ) . 8 ( 4 x ) .

Use the Associative Property of Multiplication to simplify the given expression: −9 ( 7 y ) . −9 ( 7 y ) .

Evaluate Expressions using the Commutative and Associative Properties

The commutative and associative properties can make it easier to evaluate some algebraic expressions. Since order does not matter when adding or multiplying three or more terms, we can rearrange and re-group terms to make our work easier, as the next several examples illustrate.

Example 7.8

Evaluate each expression when x = 7 8 . x = 7 8 .

Solution

What was the difference between part ⓐ and part ⓑ ? Only the order changed. By the Commutative Property of Addition, x + 0.37 + ( − x ) = x + ( − x ) + 0.37 . x + 0.37 + ( − x ) = x + ( − x ) + 0.37 . But wasn’t part ⓑ much easier?

Let’s do one more, this time with multiplication.

Example 7.9

Evaluate each expression when n = 17 . n = 17 .

Solution

What was the difference between part ⓐ and part ⓑ here? Only the grouping changed. By the Associative Property of Multiplication, 4 3 ( 3 4 n ) = ( 4 3 · 3 4 ) n . 4 3 ( 3 4 n ) = ( 4 3 · 3 4 ) n . By carefully choosing how to group the factors, we can make the work easier.

Simplify Expressions Using the Commutative and Associative Properties

When we have to simplify algebraic expressions, we can often make the work easier by applying the Commutative or Associative Property first instead of automatically following the order of operations. Notice that in Example 7.8 part ⓑ was easier to simplify than part ⓐ because the opposites were next to each other and their sum is 0 . 0 . Likewise, part ⓑ in Example 7.9 was easier, with the reciprocals grouped together, because their product is 1 . 1 . In the next few examples, we’ll use our number sense to look for ways to apply these properties to make our work easier.

Example 7.10

Simplify: −84 n + ( −73 n ) + 84 n . −84 n + ( −73 n ) + 84 n .

Solution

Notice the first and third terms are opposites, so we can use the commutative property of addition to reorder the terms.

Simplify: −27 a + ( −48 a ) + 27 a . −27 a + ( −48 a ) + 27 a .

Simplify: 39 x + ( −92 x ) + ( −39 x ) . 39 x + ( −92 x ) + ( −39 x ) .

Now we will see how recognizing reciprocals is helpful. Before multiplying left to right, look for reciprocals—their product is 1 . 1 .

Example 7.11

Simplify: 7 15 · 8 23 · 15 7 . 7 15 · 8 23 · 15 7 .

Solution

Notice the first and third terms are reciprocals, so we can use the Commutative Property of Multiplication to reorder the factors.

Simplify: 9 16 · 5 49 · 16 9 . 9 16 · 5 49 · 16 9 .

Simplify: 6 17 · 11 25 · 17 6 . 6 17 · 11 25 · 17 6 .

In expressions where we need to add or subtract three or more fractions, combine those with a common denominator first.

Example 7.12

Simplify: ( 5 13 + 3 4 ) + 1 4 . ( 5 13 + 3 4 ) + 1 4 .

Solution

Notice that the second and third terms have a common denominator, so this work will be easier if we change the grouping.

Simplify: ( 7 15 + 5 8 ) + 3 8 . ( 7 15 + 5 8 ) + 3 8 .

Simplify: ( 2 9 + 7 12 ) + 5 12 . ( 2 9 + 7 12 ) + 5 12 .

When adding and subtracting three or more terms involving decimals, look for terms that combine to give whole numbers.

Example 7.13

Simplify: ( 6.47 q + 9.99 q ) + 1.01 q . ( 6.47 q + 9.99 q ) + 1.01 q .

Solution

Notice that the sum of the second and third coefficients is a whole number.

Many people have good number sense when they deal with money. Think about adding 99 99 cents and 1 1 cent. Do you see how this applies to adding 9.99 + 1.01 ? 9.99 + 1.01 ?

Simplify: ( 5.58 c + 8.75 c ) + 1.25 c . ( 5.58 c + 8.75 c ) + 1.25 c .

Simplify: ( 8.79 d + 3.55 d ) + 5.45 d . ( 8.79 d + 3.55 d ) + 5.45 d .

No matter what you are doing, it is always a good idea to think ahead. When simplifying an expression, think about what your steps will be. The next example will show you how using the Associative Property of Multiplication can make your work easier if you plan ahead.

Example 7.14

Simplify the expression: [ 1.67 ( 8 ) ] ( 0.25 ) . [ 1.67 ( 8 ) ] ( 0.25 ) .

Solution

When simplifying expressions that contain variables, we can use the commutative and associative properties to re-order or regroup terms, as shown in the next pair of examples.

Example 7.15

Solution

In The Language of Algebra, we learned to combine like terms by rearranging an expression so the like terms were together. We simplified the expression 3 x + 7 + 4 x + 5 3 x + 7 + 4 x + 5 by rewriting it as 3 x + 4 x + 7 + 5 3 x + 4 x + 7 + 5 and then simplified it to 7 x + 12 . 7 x + 12 . We were using the Commutative Property of Addition.

Example 7.16

Simplify: 18 p + 6 q + ( −15 p ) + 5 q . 18 p + 6 q + ( −15 p ) + 5 q .

Solution

Use the Commutative Property of Addition to re-order so that like terms are together.

Simplify: 23 r + 14 s + 9 r + ( −15 s ) . 23 r + 14 s + 9 r + ( −15 s ) .

Simplify: 37 m + 21 n + 4 m + ( −15 n ) . 37 m + 21 n + 4 m + ( −15 n ) .

Links To Literacy

Section 7.2 Exercises

Practice Makes Perfect

Use the Commutative and Associative Properties

In the following exercises, use the commutative properties to rewrite the given expression.

In the following exercises, use the associative properties to rewrite the given expression.

Evaluate Expressions using the Commutative and Associative Properties

In the following exercises, evaluate each expression for the given value.

Simplify Expressions Using the Commutative and Associative Properties

In the following exercises, simplify.

3 8 g + 1 12 h + 7 8 g + 5 12 h 3 8 g + 1 12 h + 7 8 g + 5 12 h

5 6 a + 3 10 b + 1 6 a + 9 10 b 5 6 a + 3 10 b + 1 6 a + 9 10 b

Everyday Math

ⓐ How much will Allie’s stamps cost?

ⓑ How much will Loren’s stamps cost?

ⓒ What is the total cost of the girls’ stamps?

ⓐ How much money is in the first envelope?

ⓑ How much money is in the second envelope?

ⓒ What is the total value of all the cash?

ⓓ What is the value of all the $5 $5 bills?

ⓔ What is the value of all $10 $10 bills?

ⓕ What is the value of all $20 $20 bills?

Writing Exercises

In your own words, state the Commutative Property of Addition and explain why it is useful.

In your own words, state the Associative Property of Multiplication and explain why it is useful.

Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ After reviewing this checklist, what will you do to become confident for all objectives?

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    Routing

    1. Consider the network below and assume each node initially knows the costs to each of its neighbours. Consider the distance-vector algorithm and show the distance table entries at node z when the algorithm has stabilized itself.

    A: The first table below shows the initial values at z. The second table shows the contents after one hop of information received from the neighbours, the third after two hops, and the last after three hops (where the least-cost path from x-to-u via z is found).

    Page responsible: Andrei Gurtov
    Last updated: 2009-10-21

    Department of Computer and Information Science
    Linköping University
    581 83 LINKÖPING
    Tel: +46 13 28 10 00


    Treating Spaces as Blank Cells – With Helper Column

    You need to be careful when interacting with blank cells in Excel. Cells can appear blank to you, but Excel won’t treat them as blank. This can occur if the cell contains spaces, linebreaks, or other invisible characters. This is a common problem when importing data into Excel from other sources.

    If we need to treat any cells that only contain spaces the same way as if they were blank, then the formula in the previous example will not work. Notice how the SUMIFS Formula does not consider cell B9 below (” “) to be blank:

    To treat a cell containing only spaces as if it were a blank cell, we can add a helper column with the TRIM Function to remove the extra spaces from each cell’s value:


    Chapter 15: Accommodation duties and powers

    Guidance on housing authority duties and powers to secure accommodation for applicants how they arise and are brought to an end, including under the main housing duty.

    15.1 This chapter provides guidance on the housing authority’s duties to secure accommodation for applicants, how they arise and are brought to an end and the powers within the 1996 Act to secure accommodation for homeless households.

    15.2 The chapter includes:

    (a) duties to provide interim accommodation

    (b) powers to provide accommodation pending review or appeal

    (c) duties to prevent and relieve homelessness, including a power to provide accommodation

    (d) the section 193C(4) duty to secure accommodation for applicants who are homeless, eligible for assistance, have priority need and are not intentionally homeless but have deliberately and unreasonably refused to cooperate

    (e) the section 193(2) duty to secure accommodation for applicants who are homeless, eligible for assistance, have priority need and are not intentionally homeless (the main housing duty).

    Duties to provide interim accommodation

    15.3 The 1996 Act provides 4 circumstances in which a housing authority must secure accommodation on an interim basis until a decision or other event occurs. These are set out below.

    Section 188 interim duty to accommodate

    15.4 Section 188(1) requires housing authorities to secure that accommodation is available for an applicant (and their household) if they have reason to believe that the applicant may:

    (b) be eligible for assistance and,

    15.5 The threshold for triggering the section 188(1) duty is low as the housing authority only has to have a reason to believe (rather than being satisfied) that the applicant may be homeless, eligible for assistance and have a priority need.

    15.6 The section 188(1) interim accommodation duty applies even where the housing authority considers the applicant may not have a local connection with their district and may have one with the district of another housing authority giving rise to the possibility of referral (section 188(2)). For further guidance on local connection see Chapter 10.

    Ending the section 188 interim duty

    15.7 The section 188(1) interim duty comes to an end when applicants are notified of certain decisions in relation to their application.

    15.8 Following inquiries, where the housing authority concludes that an applicant does not have a priority need, the section 188(1) duty ends when either:

    (a) the housing authority notifies the applicant of the decision that they do not owe a section 189B(2) relief duty or,

    (b) the housing authority notifies them of a decision that, once the section 189B(2) relief duty comes to an end, they do not owe a duty under section 190 (duties to persons becoming homeless intentionally) or section 193(2) (the main housing duty owed to applicants with priority need who are not homeless intentionally).

    15.9 So, an applicant who the housing authority has found to be not in priority need within the 56 day ‘relief stage’ will no longer be owed a section 188(1) interim duty to accommodate, but will continue to be owed a section 189B(2) relief duty until that duty ends or is found not to be owed.

    15.10 For any other case (including for applicants who have a priority need, and for applicants who the housing authority have reason to believe will be owed a duty because they have reapplied within 2 years of accepting a private rented sector offer (for further guidance on reapplication after a private rented sector offer see Chapter 18), the section 188(1) interim duty will end at whichever is the later of:

    (a) the housing authority notifies them of what duty (if any) they are owed under Part 7 of the 1996 Act once the section 189B(2) relief duty comes to an end

    (b) the housing authority notifies them that they are not owed the section 189B(2) relief duty, or that this duty has come to an end

    (c) the housing authority notifies them of a decision following their request for a review as to the suitability of a final accommodation offer or Part 6 offer made within the section 189B relief stage.

    15.11 In summary, a housing authority may bring the section 188(1) interim accommodation duty to an end within the 56 day period (the relief stage) by issuing a section 184 decision that the applicant does not have priority need or by issuing a notification that the relief duty is not owed or has been brought to an end. If neither of these notifications is issued within the 56 day period, the section 188(1) interim accommodation duty will be brought to an end by notification of what further duties are owed, if any, under section 193 or section 190. However, in the event that the relief duty is brought to an end following refusal of a final accommodation or Part 6 offer, and the applicant requests a review as to the suitability of the accommodation offered, the section 188(1) duty will continue until a decision on the review has been notified to the applicant.

    15.12 In circumstances where an applicant is found not to be eligible for assistance, the housing authority must provide, or secure the provision of, information and advice as set out in section 179. If (section 188) interim accommodation has been provided, notice periods should take account of the needs of the applicant and the time required for them to access assistance. For households including children or particularly vulnerable adults who are owed duties under the Children Act 1989 or Care Act 2014, local authorities should consider having arrangements in place to manage a transition in responsibilities, so that there is no break in the provision of accommodation for applicants who cease to be eligible for support under the 1996 Act.

    Section 190(2): duty to provide accommodation to applicants who are intentionally homeless

    15.13 On reaching a decision that an applicant has priority need and is intentionally homeless, the housing authority must secure accommodation for a period of time that will provide a reasonable opportunity for them to find their own accommodation.

    15.14 In determining the period of time for which accommodation will be secured under section 190(2) housing authorities must consider each case on its merits. A few weeks may provide the applicant with a reasonable opportunity to secure accommodation for themselves. However, some applicants might require longer and others, particularly where the housing authority provides pro-active and effective assistance, might require less time.

    15.15 Housing authorities will need to take into account:

    (a) the particular needs and circumstances of the applicant and the resources available to them to secure accommodation. This might include any health or support needs that make it more difficult for the applicant to find and secure accommodation, as well as the support available from their family or social network

    (b) the housing circumstances in the local area, and the length of time it might reasonably take to secure accommodation. In assessing this the housing authority might reflect on the efforts previously made by both the housing authority and the applicant to relieve their homelessness, and why these had not proved successful

    (c) arrangements that have already been made by the applicant which are likely to be successful within a reasonable timescale. For example, if the applicant has secured accommodation that is not yet available to occupy or can demonstrate that accommodation will be so secured, the housing authority should consider providing section 190(2) accommodation until the applicant is able to take up the accommodation.

    Section 199A(2) and section 200(1): duties to accommodate applicants with no local connection pending outcome of referral

    15.16 If the housing authority has notified an applicant that it proposes to refer the case to another housing authority, the notifying authority has a duty under section 199A(2) (if referral is in the relief stage of an applicant who the authority has reason to believe may have a priority need) or section 200(1) (if referral is in the section 193 main housing duty stage of an applicant who has a priority need and is unintentionally homeless) to secure that accommodation is available for the applicant until they are notified of the decision whether the conditions for referral are met. At this point the duty under section 199A(2) or 200(1) will come to an end and a duty under section 189B or section 193(2) will be owed by either the notified housing authority or the notifying housing authority. For further guidance on referrals to another housing authority see Chapter 10.

    Suitability of accommodation

    15.17 Section 206(1) provides that all accommodation provided under Part 7 of the 1996 Act must be suitable for the applicant and their household, and the suitability requirements under section 210 apply. For further guidance on the suitability of accommodation see Chapter 17. Housing authorities may take into account the interim nature of a placement when assessing whether or not it is suitable as accommodation may be suitable for a few days or weeks that would not be suitable for a longer term placement.

    15.18 The applicant does not have the right to ask for a statutory review under section 202 of the housing authority’s decision as to the suitability of interim accommodation, but housing authorities are reminded that such decisions could be subject to judicial review.

    Ending interim accommodation arrangements

    15.19 When a housing authority is satisfied that they are under no further duty to secure interim accommodation or where this duty has ended, the housing authority will need to terminate the applicant’s right of occupation. In the first instance, a housing authority should look to the terms of the licence or tenancy under which interim accommodation has been provided to establish the length of the notice period required.

    15.20 Interim accommodation is usually provided under licences excluded from the requirement to issue 4 weeks written notice provided by the Protection from Eviction Act 1977. The courts have applied this principle in cases where the accommodation provided was B&B accommodation in a hotel, and where it was a self-contained flat. Consequently, housing authorities are required only to provide an applicant with reasonable notice to vacate the accommodation, and do not need to apply for a possession order from the court. However, housing authorities are public bodies and so must act reasonably by giving the applicant at least some opportunity to find alternative accommodation before the interim accommodation is terminated. What is considered ‘reasonable notice’ would depend on the facts of the case, taking into account the circumstances of the applicant and allowing time for them to consider whether to request a review of the decision.

    15.21 In cases involving applicants who have children under 18 where the housing authority have reason to believe that the applicant may be ineligible for assistance or may be homeless intentionally, the housing authority must, subject to the applicant’s consent, alert the children’s services authority to the case. A referral to the children’s services authority may also be made without the applicant’s consent where there are safeguarding concerns, in accordance with local procedures.

    Refusal or loss of interim accommodation

    15.22 Where an applicant rejects an offer of interim accommodation (or accepts and moves into the interim accommodation and then later rejects it), this will bring the housing authority’s interim accommodation duty to an end – unless it is reactivated by any change of circumstances. Note, however, that an applicant’s rejection of interim accommodation does not end other duties that the housing authority may owe under Part 7.

    Discretionary powers to secure accommodation

    15.23 Housing authorities have powers to secure accommodation for certain applicants who request a review of certain decisions on their case, and to certain applicants requesting accommodation pending determination of a county court appeal.

    15.24 The fact that a housing authority has decided that an applicant is not eligible for housing assistance under Part 7 does not preclude it from exercising its powers to secure accommodation pending a review or appeal. However, housing authorities should note that section 54 of, and schedule 3 to, the Nationality, Immigration and Asylum Act 2002 prevent them from exercising their powers to accommodate an applicant pending a review or appeal to the county court, where the applicant is a person who falls within one of a number of classes of person specified in schedule 3 unless there would otherwise be a breach of the person’s rights under the European Court of Human Rights or rights under EU Treaties (see paragraph 7.22). For further guidance on eligibility see Chapter 7.

    Powers to accommodate pending a review

    15.25 Under section 202, applicants have the right to ask for a review of a housing authority’s decision on a number of issues relating to their case, and may also request that accommodation is secured for them pending a decision on the review. For further guidance on reviews see Chapter 19. Housing authorities have powers to accommodate applicants pending a decision on reviews under section 188(3), section 199A(6) and section 200(5) of the 1996 Act.

    15.26 In considering whether to secure accommodation pending review housing authorities will need to balance the objective of maintaining fairness between homeless persons in circumstances where they have decided that no duty is owed to them, against proper consideration of the possibility that the applicant might be right. Housing authorities should consider the following, along with any other relevant factors:

    (a) the merits of the applicant’s case that the original decision was flawed and the extent to which it can properly be said that the decision was one which was either contrary to the apparent merits or was one which involved a very fine balance of judgment

    (b) whether any new material, information or argument has been put to them which could alter the original decision and,

    (c) the personal circumstances of the applicant and the consequences to them of a decision not to exercise the discretion to accommodate.

    15.27 Where an applicant is refused accommodation pending a review, they may seek to challenge the decision through judicial review.

    Power to accommodate pending an appeal to the county court

    15.28 Where an applicant is dissatisfied with a housing authority’s section 202 review decision or are not notified of the review decision within the proper time limits, an applicant has the right to appeal to the county court on a point of law arising from the review decision or original homelessness decision. For further guidance on reviews see Chapter 19. Under section 204(4), housing authorities have the power to accommodate certain applicants:

    (a) during the period for making an appeal against their decision and,

    (b) if an appeal is brought, until it and any subsequent appeals are finally determined.

    15.29 This power may be exercised where the housing authority was previously under a duty to secure accommodation for the applicant’s occupation under section 188, section 190, section 199A or section 200 and may be exercised whether or not the housing authority has exercised its powers to accommodate the applicant pending review.

    15.30 In deciding whether to exercise this power, housing authorities will need to adopt the same approach, and consider the same factors, as for a decision whether to exercise their power to accommodate pending a review (see paragraph 15.26).

    15.31 Under section 204A, applicants have a right to appeal to the county court against decisions on the use of the section 204(4) power to accommodate. This enables an appeal against decisions not to secure accommodation for them pending their main appeal, or to stop securing accommodation, or to secure accommodation for only a limited period before final determination of the main appeal by the county court).

    15.32 In deciding a section 204A appeal, if the court quashes the decision of the housing authority, it may order the authority to accommodate the applicant, but only where it is satisfied that failure to do so would substantially prejudice the applicant’s ability to pursue the main appeal on the homelessness decision. For further guidance on reviews see Chapter 19.

    Powers to secure accommodation to prevent or relieve homelessness

    15.33 Housing authorities have duties to help prevent and relieve homelessness for eligible applicants who are threatened with becoming homelessness within 56 days, or are homeless. The section 195(2) prevention duty requires authorities to take reasonable steps to help the applicant to secure that accommodation does not cease to be available to them, and the relief duty requires housing authorities to take reasonable steps to help the applicant to secure that suitable accommodation becomes available to them for at least 6 months. For further guidance on the prevention duty see Chapter 12 and for further guidance on the relief duty see Chapter 13.

    15.34 Section 205(3) of the 1996 Act enables housing authorities to discharge the section 189B(2) relief and/or section 195(2) prevention duties by securing accommodation for an applicant, where it decides to do so. The power to secure accommodation to applicants to prevent or relieve homelessness, regardless of priority need status, provides more flexibility to pursue appropriate housing options for applicants.

    15.35 Housing authorities might use the section 205(3) power to deliver accommodation services for groups that are at higher risk of homelessness, for example young people with low incomes. The power might also be used to provide additional help to those least able to secure accommodation directly from a private landlord, such as people with an offending history or people with a mental health problem. Housing authorities will wish to consider local priorities, needs and resources when considering how the power might best be utilised in their district.

    Section 193C(4): duty to accommodate applicants who have deliberately and unreasonably refused to co-operate pending final offer

    15.36 Applicants who have priority need but are no longer owed a section 189B relief duty following service of a section 193B notice due to their deliberate and unreasonable refusal to co-operate will not be owed the section 193 main housing duty but will be owed an accommodation duty under section 193C(4).

    15.37 This section 193C(4) duty ends if the applicant accepts or refuses a final accommodation offer or a final Part 6 offer. A ‘final accommodation offer’ is an offer of an assured shorthold tenancy made by a private landlord with the approval of the housing authority, with a view to bringing the section 193C(4) duty to an end. The offer must be of a fixed term tenancy (within the meaning of Part 1 of the Housing Act 1988) of at least 6 months duration, and the accommodation must be suitable for the applicant. A ‘final Part 6 offer’ is a suitable housing allocation (under Part 6 of the 1996 Act) made in writing, and which states that it is a final offer for the purposes of this section. A housing authority must not approve a final accommodation offer or make a final Part 6 offer if the applicant has a contractual obligation in respect of their existing accommodation which they are unable to bring to an end before being required to take up the offer. For further guidance on suitability see Chapter 17.

    15.38 The section 193C(4) duty will also end if the applicant:

    (a) ceases to be eligible for assistance

    (b) becomes homeless intentionally from the accommodation provided under section 193C(4)

    (c) accepts an offer of an assured tenancy from a private landlord or,

    (d) voluntarily ceases to occupy as their only or principal home, the accommodation provided.

    Duty to secure accommodation under the section 193(2) ‘main housing duty’

    15.39 Where an applicant is unintentionally homeless, eligible for assistance and has a priority need for accommodation, the housing authority has a duty under section 193(2) to secure that accommodation is available for their occupation (unless it refers the application to another housing authority under section 198). This is commonly known as ‘the main housing duty’. However, the main housing duty will not be owed to an applicant who has turned down a suitable final accommodation offer or Part 6 offer made during the section 189B(2) relief stage, or has been given notice under section 193B(2) due to their deliberate and unreasonable refusal to co-operate. For further guidance on deliberate and unreasonable refusal to co-operate see Chapter 14.

    15.40 The accommodation secured must be available for occupation by the applicant together with any other person who normally resides with them as a member of their family, or any other person who might reasonably be expected to reside with them. It must be suitable for their occupation. For further guidance on suitability see Chapter 17.

    15.41 The housing authority will cease to be subject to the duty under section 193(2) (the main housing duty) in the following circumstances:

    (a) the applicant accepts a suitable offer of accommodation under Part 6 (an allocation of social housing) (section 193(6)(c)). This would include an offer of an assured tenancy of a private registered provider property via the housing authority’s allocation scheme

    (b) the applicant accepts an offer of an assured tenancy (other than an assured shorthold tenancy) from a private landlord (section 193(6)(cc)). This could include an offer of an assured tenancy made by a private registered provider

    (c) the applicant accepts or refuses a private rented sector offer - an offer of an assured shorthold tenancy of at least 12 months made by a private landlord (section 193(7AA)). For this to be the case the applicant must have been informed in writing of the possible consequences of refusing or accepting the offer, their right to request a review of the suitability of the accommodation, and the duties that would be owed to them on re-application if they became unintentionally homeless from the accommodation within 2 years of accepting the offer

    (d) the applicant refuses a final offer of accommodation under Part 6 (an allocation of social housing). The main housing duty does not end unless the applicant is informed of the possible consequences of refusal and of their right to ask for a review of the suitability of the accommodation (section 193(7)), the offer is made in writing and states that it is a final offer (section 193(7A)), and the housing authority is satisfied that the accommodation is suitable and that it would be reasonable for the applicant to accept it (section 193(7F))

    (e) the applicant refuses an offer of temporary accommodation which the housing authority is satisfied is suitable for the applicant (section 193(5)). For this to be the case the applicant must have been informed of the possible consequences of refusal and of their right to ask for a review of the suitability of the accommodation, and have been notified by the housing authority that it regards itself as having discharged its duty.

    15.42 The main housing duty will also end if the applicant:

    (a) ceases to be eligible for assistance as defined in section 185 of the 1996 Act

    (b) becomes homeless intentionally from accommodation made available to them under section 193. For further guidance on intentional homelessness see Chapter 9

    (c) voluntarily ceases to occupy as their principal home the accommodation made available under section 193.

    15.43 In restricted cases housing authorities should, as far as is reasonably practicable, bring the section 193(2) duty to an end through the offer of a assured shorthold tenancy of at least 12 months duration with a private landlord (section 193(7AD)). The applicant will not be owed a section 195A duty if they re-apply as unintentionally homeless within 2 years of accepting the offer. For further guidance on restricted cases see Chapter 7.

    Pre-Localism Act 2011 cases

    15.44 In circumstances where the housing authority accepted a section 193(2) duty on an application made before 9 November 2012, that duty cannot be brought to an end through an offer of private rented accommodation unless that offer meets the requirements of ‘a qualifying offer’.

    15.45 A qualifying offer must be of a fixed-term tenancy of at least 12 months duration and be accompanied by a written statement that states the term of the tenancy being offered and explains in ordinary language that there is no obligation on the applicant to accept the offer, but if the offer is accepted the housing authority will cease to be subject to the section 193 duty. The applicant must have signed a statement acknowledging that he or she has understood the written statement accompanying the offer.

    Making suitable offers

    15.46 The Secretary of State recommends that applicants are given the chance to view accommodation that is offered on anything other than an interim basis, before being required to decide whether they accept or refuse an offer, and before being required to sign any written agreement relating to the accommodation (e.g. a tenancy agreement). Where housing authorities are making offers of accommodation outside their district they should take particular care to ensure that applicants have sufficient information about the location of the accommodation and the services that would be available to them there and that applicants are given a reasonable amount of time to consider the offer made before reaching a decision. Under section 202(1A), an applicant who is offered accommodation can request a review of its suitability whether or not they have accepted the offer. For further guidance on suitability see Chapter 17 and for further guidance on reviews see Chapter 19.

    15.47 Where an applicant has contractual or other obligations in respect of their existing accommodation (e.g. a tenancy agreement or lease), the housing authority can only reasonably expect an offer to be taken up if the applicant is able to bring those obligations to an end before being required to take up the offer (section 193(8)).

    15.48 Housing authorities should allow applicants a reasonable period for considering offers of accommodation that will bring the main housing duty to an end whether accepted or refused. There is no set reasonable period some applicants may require longer than others depending on their circumstances, whether they wish to seek advice in making their decision and whether they are already familiar with the property or locality in question. Longer periods may be required where the applicant is in hospital or temporarily absent from the district. In deciding what a reasonable period is, housing authorities must take into account the applicant’s circumstances.

    15.49 For further guidance on accommodation arrangements see: