# 1.8E: Exercises - Variation

M(Marecek)_Bookshelves/Algebra/Book:_Elementary_Algebra_(OpenStax)/08:_Rational_Expressions_and_Equations/8.09:_Use_Direct_and_Inverse_Variation

### A: Translate Words into a Formula

Exercise (PageIndex{A}): Translate Words into a Formula

Translate each of the following sentences into a mathematical formula.

1. The distance (D) an automobile can travel is directly proportional to the time (t) that it travels at a constant speed.

2. The extension of a hanging spring (d) is directly proportional to the weight (w) attached to it.

3. An automobile’s braking distance (d) is directly proportional to the square of the automobile’s speed (v).

4. The volume (V) of a sphere varies directly as the cube of its radius (r).

5. The volume (V) of a given mass of gas is inversely proportional to the pressure (p) exerted on it.

6. Every particle of matter in the universe attracts every other particle with a force (F) that is directly proportional to the product of the masses (m_{1}) and (m_{2}) of the particles, and it is inversely proportional to the square of the distance d between them.

7. Simple interest (I) is jointly proportional to the annual interest rate (r) and the time (t) in years a fixed amount of money is invested.

8. The time (t) it takes an object to fall is directly proportional to the square root of the distance (d) it falls.

 1. (D=kt) 3. (d=kv^{2}) 5. (V = frac{k}{p}) 7. (I=krt)

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### B: Translate Words and Find a Formula

Exercise (PageIndex{B}): Translate Words and Find a Formula

Construct a mathematical model given the following:

9. (y) varies directly as (x), and (y=30) when (x=6).

10. (y) varies directly as (x), and (y=52) when (x=4).

11. (y) is directly proportional to (x), and (y=12) when (x=3).

12. (y) is directly proportional to (x), and (y=120) when (x=20).

13. (y) is directly proportional to (x), and (y=3) when (x=9).

14. (y) is directly proportional to (x), and (y=21) when (x=3).

15. (y) varies inversely as (x), and (y=2) when (x=frac{1}{8}).

16. (y) varies inversely as (x), and (y=frac{3}{2}) when (x=frac{1}{9}).

17. (y) is jointly proportional to (x) and (z), where (y=2) when (x=1) and (z=3).

18. (y) is jointly proportional to (x) and (z), where (y=15) when (x=3) and (z=7).

19. (y) varies jointly as (x) and (z), where (y=frac{2}{3}) when (x=frac{1}{2}) and (z=12).

20. (y) varies jointly as (x) and (z), where (y=5) when (x=frac{3}{2}) and (z=frac{2}{9}).

21. (y) varies directly as the square of (x), where (y=45) when (x=3).

22. (y) varies directly as the square of (x), where (y=3) when (x=frac{1}{2}).

23. (y) is inversely proportional to the square of (x), where (y=27) when (x=frac{1}{3}).

24. (y) is inversely proportional to the square of (x), where (y=9) when (x=frac{2}{3}).

25. (y) varies jointly as (x) and the square of (z), where (y=6) when (x=frac{1}{4}) and (z=frac{2}{3}).

26. (y) varies jointly as (x) and (z) and inversely as the square of (w), where (y=5) when (z=1, z=3), and (w=frac{1}{2}).

27. (y) varies directly as the square root of (x) and inversely as the square of (z), where (y=15) when (x=25) and (z=2).

28. (y) varies directly as the square of (x) and inversely as (z) and the square of (w), where (y=14) when (x=4, w=2) and (z=2).

 9. (y=5x)11. (y=4x) 13. (y=frac{27}{x})15. (y=frac{1}{4x}) 17. (y=frac{2}{3}xz)19. (y=frac{1}{9}xz) 21. (y=5x^{2})23. (y = frac { 3 } { x ^ { 2 } }) 25. (y = 54 x z ^ { 2 })27. (y = frac { 12 sqrt { x } } { z ^ { 2 } })
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### C: Direct variation problems

Exercise (PageIndex{C}): Direct variation problems

Solve applications involving variation.

29. Revenue in dollars is directly proportional to the number of branded sweatshirts sold. The revenue earned from selling (25) sweatshirts is ($318.75). Determine the revenue if (30) sweatshirts are sold. 30. The sales tax on the purchase of a new car varies directly as the price of the car. If an ($18,000) new car is purchased, then the sales tax is ($1,350). How much sales tax is charged if the new car is priced at ($22,000)?

31. The price of a share of common stock in a company is directly proportional to the earnings per share (EPS) of the previous (12) months. If the price of a share of common stock in a company is $22.55, and the EPS is published to be ($1.10), determine the value of the stock if the EPS increases by ($0.20). 32. The distance traveled on a road trip varies directly with the time spent on the road. If a (126)-mile trip can be made in (3) hours, then what distance can be traveled in (4) hours? 33. The circumference of a circle is directly proportional to its radius. The circumference of a circle with radius (7) centimeters is measured as (14π) centimeters. What is the constant of proportionality? 34. The area of circle varies directly as the square of its radius. The area of a circle with radius (7) centimeters is determined to be (49π) square centimeters. What is the constant of proportionality? 35. The surface area of a sphere varies directly as the square of its radius. When the radius of a sphere measures (2) meters, the surface area measures (16π) square meters. Find the surface area of a sphere with radius (3) meters. 36. The volume of a sphere varies directly as the cube of its radius. When the radius of a sphere measures (3) meters, the volume is (36π) cubic meters. Find the volume of a sphere with radius (1) meter. 37. With a fixed height, the volume of a cone is directly proportional to the square of the radius at the base. When the radius at the base measures (10) centimeters, the volume is (200) cubic centimeters. Determine the volume of the cone if the radius of the base is halved. 38. The distance (d) an object in free fall drops varies directly with the square of the time (t) that it has been falling. If an object in free fall drops (36) feet in (1.5) seconds, then how far will it have fallen in (3) seconds?  Hooke’s law suggests that the extension of a hanging spring is directly proportional to the weight attached to it. The constant of variation is called the spring constant. Robert Hooke (1635-1703) 39. A hanging spring is stretched (5) inches when a (20)-pound weight is attached to it. Determine its spring constant. 40. A hanging spring is stretched (3) centimeters when a (2)-kilogram weight is attached to it. Determine the spring constant. 41. If a hanging spring is stretched (3) inches when a (2)-pound weight is attached, how far will it stretch with a (5)-pound weight attached? 42. If a hanging spring is stretched (6) centimeters when a (4)-kilogram weight is attached to it, how far will it stretch with a (2)-kilogram weight attached? The braking distance of an automobile is directly proportional to the square of its speed. 43. It takes (36) feet to stop a particular automobile moving at a speed of (30) miles per hour. How much breaking distance is required if the speed is (35) miles per hour? 44. After an accident, it was determined that it took a driver (80) feet to stop his car. In an experiment under similar conditions, it takes (45) feet to stop the car moving at a speed of (30) miles per hour. Estimate how fast the driver was moving before the accident. 45. The period (T) of a pendulum is directly proportional to the square root of its length (L). If the length of a pendulum is (1) meter, then the period is approximately (2) seconds. Approximate the period of a pendulum that is (0.5) meter in length. 46. The time (t) it takes an object to fall is directly proportional to the square root of the distance (d) it falls. An object dropped from (4) feet will take (frac{1}{2}) second to hit the ground. How long will it take an object dropped from (16) feet to hit the ground? Answers 29-45:  29. ($382.50)31. ($26.65)33. (2π) 35. (36π) square meters37. (50) cubic centimeters39. (frac{1}{4}) 41. (7.5) inches43. (49) feet45. (1.4) seconds ( igstar) ### D: Inverse Variation Problems Exercise (PageIndex{D}): Inverse Variation Problems 47. To balance a seesaw, the distance from the fulcrum that a person must sit is inversely proportional to his weight. If a (72)-pound boy is sitting (3) feet from the fulcrum, how far from the fulcrum must a (54)-pound boy sit to balance the seesaw? 48. The current (I) in an electrical conductor is inversely proportional to its resistance (R). If the current is (frac{1}{4}) ampere when the resistance is (100) ohms, what is the current when the resistance is (150) ohms? 49. The amount of illumination (I) is inversely proportional to the square of the distance (d) from a light source. If (70) foot-candles of illumination is measured (2) feet away from a lamp, what level of illumination might we expect (frac{1}{2}) foot away from the lamp? 50. If (40) foot-candles of illumination is measured (3) feet away from a lamp, at what distance can we expect (10) foot-candles of illumination?  Boyle’s law states that if the temperature remains constant, the volume (V) of a given mass of gas is inversely proportional to the pressure (p) exerted on it. Robert Boyle (1627-1691) 51. A balloon is filled to a volume of (216) cubic inches on a diving boat under (1) atmosphere of pressure. If the balloon is taken underwater approximately (33) feet, where the pressure measures (2) atmospheres, then what is the volume of the balloon? 52. A balloon is filled to (216) cubic inches under a pressure of (3) atmospheres at a depth of (66) feet. What would the volume be at the surface, where the pressure is (1) atmosphere? Answers 47-51:  47. (4) feet 49. (1,120) foot-candles 51. (108) cubic inches ( igstar) ### E: Joint and Combined Variation Exercise (PageIndex{E}): Joint and Combined Variation 53. The number of men, represented by (y), needed to lay a cobblestone driveway is directly proportional to the area (A) of the driveway and inversely proportional to the amount of time (t) allowed to complete the job. Typically, (3) men can lay (1,200) square feet of cobblestone in (4) hours. How many men will be required to lay (2,400) square feet of cobblestone in (6) hours? 54. The volume of a right circular cylinder varies jointly as the square of its radius and its height. A right circular cylinder with a (3)-centimeter radius and a height of (4) centimeters has a volume of (36π) cubic centimeters. Find a formula for the volume of a right circular cylinder in terms of its radius and height.  Newton’s universal law of gravitation states that every particle of matter in the universe attracts every other particle with a force (F) that is directly proportional to the product of the masses (m_{1}) and (m_{2}) of the particles and inversely proportional to the square of the distance (d) between them. The constant of proportionality is called the gravitational constant. Sir Isaac Newton (1643-1727) 55. If two objects with masses (50) kilograms and (100) kilograms are (frac{1}{2}) meter apart, then they produce approximately (1.34 × 10^{−6}) newtons (N) of force. Calculate the gravitational constant. 56. Use the gravitational constant from the previous exercise to write a formula that approximates the force (F) in newtons between two masses (m_{1}) and (m_{2}), expressed in kilograms, given the distance (d) between them in meters. 57. Calculate the force in newtons between Earth and the Moon, given that the mass of the Moon is approximately (7.3 × 10^{22}) kilograms, the mass of Earth is approximately (6.0 × 10^{24}) kilograms, and the distance between them is on average (1.5 × 10^{11}) meters. 58. Calculate the force in newtons between Earth and the Sun, given that the mass of the Sun is approximately (2.0 × 10^{30}) kilograms, the mass of Earth is approximately (6.0 × 10^{24}) kilograms, and the distance between them is on average (3.85 × 10^{8}) meters. 59. If (y) varies directly as the square of (x), then how does (y) change if (x) is doubled? 60. If (y) varies inversely as square of (t), then how does (y) change if (t) is doubled? 61. If (y) varies directly as the square of (x) and inversely as the square of (t), then how does (y) change if both (x) and (t) are doubled? Answers 53-61:  53. (4) men55. (6.7 imes 10 ^ { - 11 } mathrm { Nm } ^ { 2 } / mathrm { kg } ^ { 2 }) 57. (1.98 imes 10 ^ { 20 } mathrm { N })59. (y) changes by a factor of (4) 61. (y) remains unchanged ( igstar) xxx ### F: More Variation Problems Exercise (PageIndex{F}): Variation Applications Solve the following variation problems. 71. The number of calories, c, burned varies directly with the amount of time, t, spent exercising. Arnold burned 312 calories in 65 minutes exercising. How many calories would he burn if he exercises for 90 minutes? 72. The number of gallons of gas a car uses varies directly with the number of miles driven. Driving 469.8 miles used 14.5 gallons of gas. How many gallons of gas would the car use if driven 1000 miles? 73. The weight of a liquid varies directly as its volume. A liquid that weighs 24 pounds has a volume of 4 gallons. If a liquid has volume 13 gallons, what is its weight? 74. The maximum load a beam will support varies directly with the square of the diagonal of the beam’s cross-section. A beam with diagonal 4” will support a maximum load of 75 pounds. What is the maximum load that can be supported by a beam with diagonal 8”? 75. The area of a circle varies directly as the square of the radius. A circular pizza with a radius of 6 inches has an area of 113.04 square inches. What is the area of a pizza with a radius of 9 inches? 76. The fuel consumption (mpg) of a car varies inversely with its weight. A car that weighs 3100 pounds gets 26 mpg on the highway. What would be the fuel consumption of a car that weighs 4030 pounds? 77. A car’s value varies inversely with its age. If a two-year-old car is worth$20,000, what will be the value of the car when it is 5 years old?

78. The number of hours it takes for ice to melt varies inversely with the air temperature. Suppose a block of ice melts in 2 hours when the temperature is 65 degrees. How many hours would it take for the same block of ice to melt if the temperature was 78 degrees?

79. The force needed to break a board varies inversely with its length. Richard uses 24 pounds of pressure to break a 2-foot long board. How many pounds of pressure is needed to break a 5-foot long board?

80. For people with roughly the same build, the weight of the person varies as the cube of their height. If a person 65 inches high weighs 125 pounds, how much would a person 75 inches high with a similar build be expected to weigh?

81. The fuel consumption (mpg) of a car varies inversely with its weight. A Ford Focus weighs 3000 pounds and gets 28.7 mpg on the highway. What would the fuel consumption be for a Ford Expedition that weighs 5,500 pounds? Round to the nearest tenth.

82. A person's BMI (body mass index) varies directly as their weight and inversely as the square of their height. Given a person who weight 180 pounds and is 60 inches tall has a BMI of 35.2, what is the BMI for someone who is 150 pounds and 68 inches tall?

83. The maximum load (L) that a cylindrical column with a circular cross section can hold varies directly as the fourth power of the diameter (d) and inversely as the square of the height (h). If an 8.0 m column that is 2.0 m in diameter will support 64 tons, how many tons can be supported by a column 12.0 m high and 3.0 m in diameter?

84. The heat loss per hour through a glass window varies directly with the difference in temperature between the inside and outside temperatures and inversely as the thickness of the glass. A 0.3 cm thick window loses 2.4 BTU per hour when the outside temperature is 50 degrees Fahrenheit and the inside temperature is 70 degrees Fahrenheit. What will the heat loss be for a 1.5 cm thick window when the outside temperature is 30 degrees Fahrenheit and the inside temperature is 70 degrees Fahrenheit?

85. The heat loss of a glass window varies jointly as the window's area and the difference between the outside and inside temperatures. A window 3 feet wide by 6 feet long loses 1200 Btu per hour when the temperature outside is 20 degrees colder than the temperature inside. Find the heat loss through a glass window that is 6 feet wide by 9 feet long when the temperature outside is 10 degrees colder that the temperature inside.

86. Sound intensity varies inversely as the square root of the distance from the sound source. If you are in a movie theater and you change your seat to one that is twice as far from the speakers, how does the new sound intensity compare to that of your original seat?

87. The number of hours (h) that it takes (p) people to assemble (m) machines varies directly as the number of machines and inversely as the number of people. If four people can assemble 12 machines in four hours, how many people are needed to assemble 36 machines in eight hours?

88. The amount of time t needed to build a wall varies directly as the number of bricks b need and inversely as the number of workers w. If it takes 18 hours for six workers to make a wall composed of 2400 bricks, how long would it take to build a wall of 4500 bricks with 10 workers?

 71. 432 calories 73. 78 pounds 75. 254.34 square inches 77. \$8,000 79. 9.6 pounds 81. 15.6 mpg 83. 144 tons 85. 1800 BTU per hour 87. 6 people

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## 17.4: Titrations and pH Curves

In an acid&ndashbase titration, a buret is used to deliver measured volumes of an acid or a base solution of known concentration (the titrant) to a flask that contains a solution of a base or an acid, respectively, of unknown concentration (the unknown). If the concentration of the titrant is known, then the concentration of the unknown can be determined. The following discussion focuses on the pH changes that occur during an acid&ndashbase titration. Plotting the pH of the solution in the flask against the amount of acid or base added produces a titration curve. The shape of the curve provides important information about what is occurring in solution during the titration.

## Dumbbell Row Variations

How it works: For each exercise variation, do 3 sets of 10 reps.

### Step 1: Bent-Over Row

In order to master the advanced rowing exercise variations, you&aposll begin with the bent-over dumbbell row, which will help you lock down the proper form. The starting position for this move is similar to that of a deadlift, with hinged hips and a flat back.

A. Holding a dumbbell in each hand, stand with feet hip-width apart. Bend knees softly.

B. Hinge at the hips until chest is parallel with the floor and back is flat. Lower dumbbells to the floor until arms are straight.

C. Pull shoulders down and away from ears. Squeeze shoulder blades together, brace your core, draw belly button toward the spine, and slowly bend elbows to pull dumbbells back toward your hips. Keep elbows tight next to your body. Return to start.

Repeat for 10 reps.

### Step 2: Single-Leg Bent Over Row

This exercise variation is all about balance, making it a great option if you want to work on strengthening your core and its stabilizing muscles. You’ll engage your core while rowing both arms or one at a time. To get the most out of each rep, focus on keeping your hips even and parallel to the ground. (P.S. This is why you need to add balance training to your workout routine.)

A. Holding a dumbbell in each hand, stand with feet hip-width apart. Bend knees softly.

B. Hinge at the hips until chest is parallel or near parallel with the floor and back is flat. Lift one foot off the ground and extend it straight backward. Lower dumbbells to the floor until elbows are completely straight.

C. Pull shoulders down and away from ears. Squeeze shoulder blades together, tighten your core, draw belly button toward your spine, and slowly bend elbows to pull dumbbells back toward your hips. Keep elbows tight next to your body. Return to start.

Repeat for 10 reps.

### Step 3: Plank Row

The plank row is the toughest of these three rowing variations, but thanks to your work on the previous two moves, you should be ready to advance successfully. Before you begin, make sure you’ve effectively nailed proper form in a high plank position to avoid swaying your hips when you pull the dumbbell to your chest. This will ensure every rep is as effective (and injury-free) as possible.

A. Place two dumbbells shoulder-width apart on the ground. Start in high-plank position, with shoulders stacked over hands, each of which is gripping a dumbbell, feet hip-width apart.

B. Engage quads, glutes, and core, and tuck tailbone slightly. Pull shoulders down and away from ears, and slowly bend one elbow back, keeping arm close to your side as you pull dumbbell up toward your hip. Return to start, placing dumbbell down softly.

Repeat for 10 reps each side, switching sides after every rep.

## 20 Glute Bridge Variations

With all of these bridge variations, it is very important that you focus on working your glutes. You don’t want to feel your low back working. Your glutes should drive the movement.

If you do feel your low back taking over, make sure that you aren’t hyperextending your low back just to get your hips up higher. You want to fully extend your hips, but it doesn’t matter how high up you bridge if it means arching your back.

To help protect your low back, you may even want to perform a Pelvic Tilt before bridging up. For Pelvic Tilt instructions, check out this post on the Pelvic Tilt Progression.

1. Basic Glute Bridge Hold –If you’ve never done a glute bridge before, you need to start with the basic glute bridge hold. This is also a good recovery and activation move for even the most advanced lifter.

Basically EVERYONE at some point needs to do the Basic Glute Bridge Hold.

Then bend your elbows to 90 degrees so that only your upper arms are on the ground. Drive up through your heels and upper back and arm to lift your glutes up off the ground. Drive your hips up as high as possible, squeezing the glutes hard. Keep your belly button drawn in so you don’t hyperextend your back.

Do not push backward off your heels. Make sure you are driving straight up almost as if driving your knees forward over your toes. Also, make sure that your knees aren’t caving in or falling open.

Squeeze your glutes and hold at the top. Concentrate on feeling your glutes work. Don’t just go through the motions. Actually think about the muscles that should be working!

If you want to make the Basic Glute Bridge Hold more challenging, you could hold the bridge with your feet in a Power Wheel.

The instability of the wheel will make the move harder. Only add in the wheel if you can fully extend your hips and feel your glutes working with the Basic Hold and even the Single Leg Hold.

2. Single-Leg Glute Bridge Hold –By making the Glute Bridge a unilateral movement (aka holding on one leg), you will make the move more challenging since the single leg will have to do the work of both legs. Make sure you only progress to this move if you can still fully extend your hips. If you struggle to feel your glutes working on the two-leg hold, do not yet attempt this single leg variation.

To do the Single-Leg Glute Bridge Hold, set up like you would for the glute bridge and then raise one leg up off the ground. You can bend the raised leg to 90 degrees or you can straighten the leg up straight toward the ceiling. Either is fine. Just make sure you don’t swing the raised leg to help you bridge up.

Drive your hips up, pressing through your heel and upper back. Hold at the top. Keep your abs engaged so you don’t feel it in your low back. Like with all the bridge exercises, focus on feeling your glutes work and not just going through the motions of bridging up.

3. Basic Glute Bridge Reps –The Basic Glute Bridge done for reps is another great beginner bridge variation. It is also a great warm up move to get your glutes ready and activated for your leg workout.

Bend your elbows to 90 degrees so that only your upper arms are on the ground.

Then drive up through your heels and upper back to lift your glutes off the ground. Drive your hips up as high as possible, squeezing the glutes hard. Keep your belly button drawn in so you don’t hyperextend your back. Focus and consciously squeeze your glutes at the top.

Do not push backward off your heels. Make sure you are driving straight up and that your knees aren’t caving in.

Squeeze your glutes for second or two at the top and lower all the way back down to the ground before repeating.

Do not rush through the move. To make the move harder, hold longer at the top or even slow down the lower back down to the ground. A slower tempo means more time under tension and more work for your glutes.

4. Single Leg Glute Bridge –Just like with the hold, using a single leg makes the repetitions harder. Do not rush through the move just because you are using a single leg. You want to still focus on really squeezing the glutes at the top and fully extending the hips.

By doing the unilateral bridge, you force each leg to work independently, which can also help correct any imbalances you may have. Often one leg is weaker and when we do bilateral moves, the stronger leg can take over and perpetuate the imbalance. With unilateral moves, each leg is forced to work separately, which can help correct the imbalance.

To do the Single Leg Glute Bridge with reps, you will set up like you did for the Basic Glute Bridge with reps except you will raise one foot up toward the ceiling.

Bridge up, driving through your heel and upper back and arms. Hold for a second or two at the top with your hips fully extended. Then lower back down. Make sure to squeeze your glutes at the top and not hyperextend your low back just to get higher up off the ground.

Complete all reps on one side before switching. Do not alternate sides. You want to fatigue the leg before switching to the other side.

5. Mini Band Glute Bridge –If you have knee valgus when you run or squat (aka if your knees cave in), you should try the Mini Band Glute Bridge. This is a great move to include in your warm up or recovery routine especially if you are a runner.

The Mini Band Glute Bridge can be done as either a two-leg glute bridge or a single leg glute bridge. Do not do the single leg variation if you haven’t mastered the two-leg.

To do the Two-Leg Mini Band Glute Bridge, place the band around your knees or right above your knees.

Lie flat on your back with your knees bent and feet flat on the ground. Bring your heels in toward your butt. The closer to your butt your heels are, the more you will isolate the glutes. The farther your heels are from your butt, the more you will work your hamstrings.

Then bend your elbows to 90 degrees and, driving through your heels and upper back and arms, bridge up. Fully extend your hips and squeeze your glutes.

Hold for a second or two at the top with your glutes and abs engaged and then lower back down and repeat.

To do the Single-Leg Mini Band Glute Bridge, set up like you would for the two-leg variation. Then lift one foot up off the ground and extend the leg in front of you so that your quads are next to each other.

Keep tension on the band as you bridge, driving through your heel on the ground and your upper back. Do not let your knee cave in as you bridge up.

Hold for a second or two at the top and fully extend your hips, squeezing your glutes. Lower back down and repeat.

Just like with the two leg variation, if you move your heel further from your butt, you will work your hamstring more.

Make sure you do not feel either bridge variation in your low back. Only progress to the single leg variation if you can get your hips up just as high without engaging your low back.

6. Glute Bridge to Sit Up –Want to work your entire core? Then the Glute Bridge to Sit Up is a great move to include in your workout.

Make sure you don’t rush this move. You want to hold the glute bridge for a second or two before performing the sit up. Beginners may need to straighten their legs all the way out to make the sit up easier.

To do the Glute Bridge to Sit Up, lie on your back with your knees bent and feet flat on the ground about hip-width apart. Bridge up, lifting your hips as high as possible while driving through your heels. Lower your hips back down and let your feet move a little away from your butt.

Then sit up, keeping your torso up nice and tall and your feet on the ground. Once you sit all the way up, lie back down, bring your feet back in and then bridge back up.

7. Glute Bridge Off Box –If you don’t have weights, a great way to make the basic bridge more challenging is by putting your feet up on a box. This move is a great supplemental exercise to include after you’ve done your big lift, whether that big lift is a deadlift, squat or Barbell Hip Thruster.

Make sure to keep the tempo slow when doing the off box variation. Also make sure you do not feel your low back.

To do the Glute Bridge Off Box, place your heels up on the box. You want your knees bent to 90 degrees or your butt to be even closer to the box. Do not let your butt get too far away from the box.

Bend your arms to 90 degrees with just your upper arms on the ground.

Then driving through your heels on top of the box, press your hips up as high as you can. Squeeze your glutes and keep your core tight as you bridge up so that you don’t hyperextend your low back.

Also, do not let your knees fall in or out as you lift up. Keep your knees in line with your hips and ankles.

Hold for a second or two and then lower down.

This move can also be made harder by slowing down the tempo. And just like the glute bridge from the floor, it can also be progressed to a Single Leg Variation.

8. Single Leg Glute Bridge Off Box –Once the Glute Bridge Off Box is no longer challenging, you will want to try the single leg variation. Make sure that you can get your hips up just as high with the single leg variation. If you can’t fully extend your hips, regress the movement.

To do the Single Leg Glute Bridge Off the Box, place your heels on top of the box. Your butt should be right against the box or no further away then when your knees are bent to 90 degrees with your heels on the box.

Bend your arms to 90 degrees with just your upper arms on the ground.

Lift one leg up and either lift the foot straight up toward the ceiling or bend the knee to 90 degrees. Do not let the lifted leg swing to help you bridge up.

Then, drive through the heel on top of the box and press your hips up as high as you can. Squeeze your glutes and keep your core tight as you bridge up. Keep the ankle, knee and hip of the working leg in line.

If you feel this in your low back or can’t get your hips up as high as with the Two-Leg Glute Bridge Off Box, you may need to regress the movement.

Hold for a second or two and then lower down. Complete all reps on one side before switching.

Again, make sure you drive through your heels and upper back to get your hips up just as high as you do with the two-leg bridge off box. And do not use your lifted leg to help you bridge up by swinging it.

9. Mini Band Hip Thrusters –Another way to advance the basic bridge is with a Mini Band. The Mini Band adds resistance to the move and is a great tool for anyone that travels often or works out at home because it is easily transportable.

To do the Mini Band Hip Thruster, place the mini band around your hips and lie on your back on the ground. Place your feet flat on the ground. The closer your heels are to your glutes, the more the move will isolate your butt. The farther from your glutes your heels are, the more your hamstrings will be involved.

Make sure that no matter how close or far your heels are from your butt, your heels stay down. Extend your arms down by your sides and grab the back of the band behind your butt in both hands.

Press the band down into the ground and then bridge your hips up toward the ceiling, pressing against the band.

Bridge your hips up as high as you can, hold for a second and then lower down.

Do not hyperextend your low back just to bridge up higher. Make sure to drive straight up as you bridge and not push yourself backward.

Slow down the tempo of the move to make it harder or use a heavier resistance. Make sure to keep the mini band against the ground as you bridge up.

10. Weighted Glute Bridge –One of the best weighted exercises to really strengthen your glutes is the Weighted Glute Bridge or Barbell Glute Bridge. This move isolates your backside like none other to create great glute strength.

To do the Weighted Glute Bridge, sit on the ground and roll or place the barbell over your hips. Then lie back and bend your knees with your heels close to your butt.

Holding on to the barbell, drive your hips up and squeeze your glutes. Make sure you drive through your heels and upper back to lift straight up.

Do not let your knees fall open and do not hyperextend your low back as you squeeze your glutes at the top.

Hold for a second or two and lower back down.

As you lift, you may find you want to press the barbell down and away onto your thighs to help you squeeze your glutes at the top and keep your core engaged.

11. Thrusters Off Box –If you want to progress the Glute Bridge Off Box, try the Thrusters Off Box. With this move, your back and feet are raised allowing you to perform a bigger range of motion.

Thrusters are one of the best glute exercises out there and should be included in your glute training program.

Do not attempt this move if you feel the other bridge variations in your low back.

To do Thrusters Off Box, place a box and a bench close enough together that your back can be on the bench and your feet up on the box with your knees bent to about 90 degrees.

Lift your hips to full extension, hold for a second and lower back down.

Like with all the glute bridges and hip thrusters, make sure you don’t hyperextend your low back at the top. Consciously squeeze your glutes and drive up through your heels.

Make sure your knees don’t fall apart at the top but stay in line with your hips and ankles.

Drop back down so your butt goes below the height of the box and repeat. You do not have to touch the ground each time, but you do want to do a bigger range of motion than you could do from the ground. If you can’t lower down past the box, you may want to regress the move to a glute bridge variation.

12. Single Leg Thrusters Off Box –When you don’t have weight, also turn to the single leg variation to make the movement harder. Like all single leg variations, the Single Leg Thrusters Off Box, allow you to correct imbalances while also challenging each leg to work harder.

Just like with the Thruster Off Box, for the Single Leg Thrusters Off Box, your feet and upper back will both be on benches or boxes.

Once you are set with your heels on a box and your upper back on a bench, raise one heel up off the box.

Then, driving through the heel on the box and your upper back, press the toe of the raised leg up toward the ceiling while raising your hips as high as you can.

Hold at the top and then lower back down past the box and repeat. Complete all reps on one side before switching.

This move can be made even harder by slowing down the tempo and even holding longer at the top. This is a great option if you don’t have weights and really want to work your glutes at home or while traveling!

13. Barbell Hip Thrusters –The Barbell Hip Thrusters, like the Weighted Glute Bridges, are essential lifts to include in your glute workouts if you want to develop great glute power and strength.

Only progress to the Barbell Hip Thrusters after increasing your weight on the Weighted Glute Bridges. The Barbell Hip Thruster is a bigger range of motion than the Weighted Glute Bridge, which makes it easier to load your low back. Make sure to brace your abs and use your glutes to lift during the Thrusters.

If you feel it in your low back, regress to another Thruster variation or back to the Weighted Glute Bridge.

To do the Barbell Hip Thruster, set up a bench and make sure it won’t move as you bridge up onto it. Take a barbell and put some padding around it so that it won’t dig into your hips as you perform the move.

Place your upper back on the bench and sit your butt on the ground with your legs out straight. Place the barbell over your hips and plant your feet firmly on the ground and close to your butt.

Then drive up through your heels and your upper back on the bench, to lift your glutes and the barbell up toward the ceiling.

Squeeze your glutes and press your hips up as high as possible, driving the barbell up and off the ground. Hold a second or two at the top and then lower back down and repeat.

Do not hyperextend your back at the top. Keep your belly button pulled in toward your spine and really squeeze your glutes at the top. Make sure you are driving straight up through your heels. Do not push yourself backward over the bench.

You may find that as you bridge up you want to “push” the barbell down toward your thighs to help you drive up and squeeze your glutes.

14. Straight Leg Glute Bridge –Many people because they sit all day, especially women, are quad dominant. That is why it is important to not only work your glutes but also your hamstrings. The Straight Leg Glute Bridge is a great exercise to not only work your glutes, but also work your hamstrings.

To do the Straight Leg Bridge, place your heels in the Suspension Trainer foot straps. Lie back on the ground with your legs out straight in front of you. You can either bend your elbows to 90 degrees or leave your arms on the ground down by your sides.

Squeezing your glutes and driving down through your heels, lift your hips up as high as you can while keeping your legs straight.

Keep your abs engaged so that you don’t hyperextend your low back as you lift. Hold at the top for a second or two and lower back down.

Lift as high as you can without feeling your low back work. Really press your heels down in the straps as you lift.

You can lift the straps up higher or slow down the tempo of the move to make the move harder.

15. Glute Cherry Bomb –The Glute Cherry Bomb is another glute bridge exercise that also works the hamstrings. However, this move also works your glute medius and abductor muscles.

To do the Glute Cherry Bomb, place your heels in the suspension trainer straps and lie on your back on the ground.

Do not worry about how high you bridge up. This straight leg bridge won’t have a very big range of motion.

Then curl your heels in toward your butt, bridging up higher as you curl. Keep your glute engaged. You want your bridge at the end of the curl to look like a basic glute bridge.

Straighten your legs back out then and, keeping the nice straight bridge, move both feet out laterally, opening your legs up.

This is an advanced movement. Beginners may not want to include the curl and instead do a Straight Leg Glute Bridge to the “Cherry Bomb” portion (aka moving their legs open and closed).

16. Glute Bridge and Curl –The Glute Bridge and Curl is a very tough hamstrings exercise. It can be done using Valslides or a Power Wheel or Suspension Trainer.

For all the variations, check out this post – The Glute Bridge and Curl!

17. Camel –Glute bridge variations not only work the glutes but also focus on stretching the hips. And with the Camel, you not only stretch your hips and activate your glutes, but also open up your chest and shoulders after sitting hunched over a computer all day.

This is even a great stretch you can do right at your desk!

To do the Camel, start by kneeling on both legs with your feet flexed. Sit back and place your hands on your heels.

Hold for 1-2 seconds and relax back down. Repeat.

If you are less flexible, do this stretch with a couch, chair or table behind you. Kneel down and place your hands back behind you on the couch or table.

Press your chest out and arch as much as possible away from the piece behind you while leaning your head back.

Then relax back down and repeat.

18. Table Top Bridge –Like the Camel bridge, the Table Top Bridge is a great way to stretch your chest and hips while activating your glutes. This is a great move to include in your warm up routine.

To do the Table Top Bridge, start seated on the ground with your feet flat on the ground in front of you and your hands on the ground behind you.

Squeeze your glutes and lift your hips up as high as you can. Lean your head back and press your chest out as you bridge up. Really feel a nice stretch across your chest and shoulders.

Keep your core tight as you bridge up. Try to create a “table” with your body.

Drive your hips up as high as you can without hyperextending your low back. Really squeeze your glutes at the top.

Hold for a few seconds at the top then lower back down and repeat.

Keep the core engaged and even do a pelvic tilt at the top so that you engage your glutes without hyperextending your low back.

19. Sit Thru to Thoracic Bridge –The Sit Thru to Thoracic Bridge is another must-do activation move. It not only stretches your hips and activates your glutes but also stretches your thoracic spine, which most of us need after sitting hunched over all day.

Then lift your right hand and bring your left leg through and place your left foot flat on the ground. Rotate your hips up toward the ceiling, squeezing your glutes to lift them up as high as you can.

Feel your glutes working to keep BOTH hips up, while you reach your hand down. We have a tendency when we rotate to drop the hip on the side we are rotating toward. Make sure both hips stay up as high as possible.

You should feel a nice rotation and stretch through your spine. You are twisting almost like someone wringing out a towel.

Then drop your hips and step your foot back through while placing your hand back down on the ground.

Step through and rotate to the other side, bridging your hips up as high as you can as you reach toward the ground.

20. Posterior Plank –The Posterior Plank is a great way to advance the Table Top Bridge and get your hamstrings working as well as your glutes and your back. This move also stretches your chest and your hips, but it is more advanced because your legs are out straight.

Beginners may need to stick with the Table Top Bridge.

To do the Posterior Plank, start seated on the ground with your legs out straight in front of you and your hands on the ground behind your butt. Your feet should be together and your fingertips should be pointing toward your butt or out to the side.

Drive through your hands and heels and raise your hips up off the ground toward the ceiling, keeping your legs out straight.

Keep your legs straight as you bridge up and relax your head back. Your body should be in a nice straight line at the top.

Keep your core engaged and squeeze your glutes. Hold at the top then lower down and repeat.

Slow down the movement and add in a longer hold at the top of the movement to make the move more challenging. You can also lift one leg and do a Single Leg Posterior Plank to make the move harder.

These 20 Glute Bridge Variations will activate your glutes to make them stronger so you can lift more and run faster. These moves, because they are the opposite of what you do all day seated at a desk hunched over a computer, will also help you alleviate and prevent low back, hip and knee pain.

No matter which variation you choose, you need to include Glute Bridges in your leg workouts!

Want more Glute Exercises and 28 days of Booty Burner Workouts!? Check out my 28-Day Booty Burner Program!

## How to Use Pushups to Hit Every Single Muscle Group

Jeff Cavaliere of Athlean-X shares 18 pushup variations to hit every single muscle group.

Jeff Cavaliere C.S.C.S. of Athlean-X is not letting the quarantine prevent him from advocating for efficient, effective workouts. He's shared 29 of his home hacks to make your house a home gym and his favorite full body no-equipment workout&mdashwhile warning his viewers about the most common home workout mistakes you can make for good measure.

But you're not always going to have the energy to make your own equipment at home, or even to do more than just one type of movement. For everyone looking for the simplest way to work their whole body, Cavaliere has honed in on one basic movement: the pushup. He offers 18 variations that will help to make sure that you're hitting every muscle in your body, no equipment necessary.

"I promise you, by the end of this video, you'll pick up two or three variations you've never thought of before that you're going to want to start doing," Cavaliere promises. Before you get going, make sure you have the standard variation of the move perfected. Check out this guide to get it right.

#### Variation 1: Squeeze-Ups

This variation requires you to squeeze your body off the ground, not just push it off.

"If you apply that inward force on the ground towards each other, you'll get that isometric contraction of the chest," Cavaliere says.

#### Variation 2: Rotating Pushup

This variation activates the pecs.

"This gives us relative abduction. In other words, I'm not taking my arm across my body, but I can turn my body in space to create that abduction," he says.

#### Variation 3: Pancake Pushup

This move will maximize the use of your triceps.

"Essentially what you're trying to do is a tricep pushdown with a straight bar, except you don't have a tricep pushdown machine or a straight bar," Cavaliere advises. "All I'm trying to do is extend through the elbows."

#### Variation 4: Cobra Pushup

This version hits the long head of the tricep.

"As you start with elbows hiked to the side and you come up and shift your chest forward, you can see the long head get into a complete contraction."

#### Variation 5: Modified Planche Pushup

This will put the focus on developing your shoulders.

"We're getting our arms and hand position way back, as far back without tilting forward," he says. "What that does is it shifts that load away from the chest, more to that front delt."

#### Variation 6: Pushaway Pushup

This will also challenge your shoulders.

"Instead of pushing straight up and down with our chest doing most of the work, we can make our shoulders do more of the work by pushing away," he says.

#### Variation 7: Pike Pushup

We're not done with the shoulders yet. This takes the Pushaway Pushup to another level.

"With a pike you're getting more of a vertical position," he says.

#### Variation 8: Handstand Pushup

Shoulder crushing continues&mdashthis move takes the Pike Pushup one step further.

"You go completely vertical against the force of gravity, and would be like doing a standard overhead press with a barbell," he says.

#### Variation 9: Bodyweight Side Lateral Raise Pushup

This move hits the middle delt, and simulates the action of a side lateral raise.

"This is relative abduction at the shoulder," says Cavaliere.

#### Variation 10: Sliding Pulldown Pushup

Now, Cavaliere moves into moves that will target your back and lats.

Laying face down on a floor that has a slick surface for you to slide on, pull your body across the floor by abducting your elbows tight and into your sides. At the top of the pull, perform a pushup and then smoothly slide your body back out until you get a great stretch on your lats.

#### Variation 11: Thumbs Up Pushup

"What you're doing here is you're trying to drive your fists with your thumbs up, and your forearms down into the ground," Cavaliere says. "They're in this 'W' position to try to get into a more externally rotated position to protect your shoulder, driving through protraction of the shoulder blades."

#### Variation 12: Back Widow Pushup

This is a flipped-over version of the pushup (face-up) that hits the upper back.

"The upper back muscles including the rhomboids, upper traps and posterior deltoid are hit hard by this amazing posterior chain pushup version," he says.

#### Variation 13: Glute Ham Raise Pushup

Next, Cavaliere offers variations to hit more of your legs.

"Here you are going to eccentrically contract the glutes and hamstrings to lower yourself to the ground and utilize the push to get yourself back up to the halfway point, at which point your glutes and hams can contract to power you back to the top," he says.

#### Variation 14: Ham Curl Pushup

"You anchor your feet, and as we use our hands to push up off the ground, we're focusing on allowing our hamstrings to dig backwards into the ground to assist us in getting our body up," Cavaliere says. "But it's not just hitting the hamstrings. As I get up to the top, you can see all the posterior muscles are working."

#### Variation 15: Rolling Squat Pushup

"This serves more as a good combination movement, allowing us to engage those muscles," according to Cavaliere.

#### Variation 16: TKE Pushup

"You anchor your feet on something higher. As I'm pushing out of this pushup, I'm allowing the legs to drive me, forcing my knees to extend, and we know that knee extension causes quad contraction," Cavaliere instructs.

#### Variation 17: Cliffhanger Pushup

Finally, Cavaliere shows off some moves that will more directly target your abs. Remember, for many of these, you should be bracing your core to maintain proper form from the start.

"You extend your hands out in front of us, with our fingers down into the ground. Once our hands pass the level of our heads, we feel our entire front side core musculature on fire," he says.

#### Variation 18: Black Widow Knee Slide Pushups

"We perform our pushup, and as we come up, we get this bottom-up rotation of the knee heading towards the opposite elbow," Cavaliere says. "In order to get that core activated even more, we have to drive and slide that knee up as high as we can."

## One-Arm Dumbbell Row

Defining Difference: The key here is isolation. Not only will the dumbbell provide maximum range of motion, but pulling with one arm at a time also allows you to focus on bringing up the weaker side if you have an imbalance issue. Plus, because you’re using the bench for support (by having a knee on it), you don’t have to be as mindful of torso position as with standing rows you can just focus on pulling heavy weight with minimal lower-back injury risk.

Action: Place one bent knee and same-side hand on a bench with the opposite foot on the floor and that hand holding a dumbbell. Begin bent over at the waist with your back level with the floor, your head pointed down and the dumbbell hanging straight toward the floor with your arm extended, palm facing in. Keeping your chest pointed to the floor, pull the dumbbell up to your waist by contracting your back muscles and bending your elbow. When it reaches the top, squeeze your shoulder blades together, then slowly lower the dumbbell to the start position. Complete all reps with one arm, then switch arms.

When to Do It: One-arm rows can fall anywhere in your back workout—as the first exercise going heavy, or as a finisher with lighter weight and high reps.

## Fundamentals of Nursing Nursing Test Bank

In this section are the practice quiz for fundamentals of nursing that can help you think critically and augment your review for the NCLEX. There are 600+ NCLEX-style practice questions partitioned into four sets in this nursing test bank. We’ve made a significant effort to provide you with the most informative rationale so please be sure to read them. Use these nursing practice questions as an alternative for Quizlet or ATI.

Quizzes included in this guide are:

• Since this is a review, answers and rationales are shown after you click on the "Check" button.
• There is no time limit, answer the questions at your own pace.
• Once all questions are answered, you'll be prompted to click the "Quiz Summary" button where you'll be shown the questions you've answered or placed under "Review". Click on the "Finish Quiz" button to show your rating.
• After the quiz, please make sure to read the questions and rationales again by click on the "View Questions" button.
• Comment us your thoughts, scores, ratings, and questions about the quiz in the comments section below!

### Fundamentals of Nursing NCLEX Practice Questions Quiz #1: 75 Questions

This is the first set of nursing practice questions about fundamentals of nursing. Nursing topics in this set include constipation, bowel elimination,

## 12 Best Push-up Variations for Getting Toned, Strong Arms

From incline push-ups to diamond push-ups, no muscle goes unchallenged.

When you think about the most effective bodyweight exercises, push-ups should come to mind. As a total-body exercise, push-ups challenge your entire upper body, core, glutes, and legs. That's right&mdashit takes every major muscle group to lower and lift your body from the ground.

"They're [push-ups] great for building strength without equipment. In a traditional wide grip, the chest is the major muscle group worked. If done properly, push-ups are also a really great core exercise because you have to keep your abs and back tight and supported through the whole repetition," explains Larysa DiDio, a certified personal trainer and creator of Tone Up in 15, an exercise DVD with five 15-minute strength training workouts that target different areas of the body .

No matter your fitness level, you can easily intensify or dial down a push-up by changing the placement of your hands, lifting your hands or feet, and incorporating more movement in your legs. If you're a beginner, for instance, you want to start with a modified push-up on your knees until you're able to build greater upper body and core strength. And if you're a seasoned athlete, maybe you'll challenge your balance and core stability by incorporating shoulder taps.

"As you progress in push-up variations, strength is increased through increased core engagement, increased engagement of minor muscle groups&mdashnamely triceps&mdashand the introduction of balance," DiDio says. " The more unstable the push-up, the harder it is to perform, and the more you have to draw on increased strength," she adds.

To ensure you're challenging your body in new ways, we put together the best push-up variations to try as you progress. Incorporate a variety of them into your current workout routine for eight to 12 reps each. "Consistency is key. The more consistent you are in doing them, the stronger you&rsquoll get," DiDio says.

## 1. WEIGHTED SIT UP

1. Lie on your back with your knees bent and pointed to the ceiling. Hold a weight at the center of your chest (but not resting on your chest).
2. Curl up without letting your chin touch your chest. The weight might move forward (toward your midsection), but make sure that you’re holding it above you the entire time so that you feel the full weight.
3. Lower back down with control.

Start with a 5-pound medicine ball or dumbbell and work up from there.

## Study Questions:

8.1 What information, and what reagents would you need to use PCR to detect HIV in a blood sample?

8.2 A 6.0 kbp PCR fragment flanked by recognition sites for the HindIII restriction enzyme is cut with HindIII then ligated with a 3kb plasmid vector that has also been cut with HindIII. This recombinant plasmid is transformed into E. coli. From one colony a plasmid is prepared and digested with HindIII.

a) When the product of the HindIII digestion is analyzed by gel electrophoresis, what will be the size of the bands observed?

b) What bands would be observed if the recombinant plasmid was cut with EcoRI, which has only one site, directly in the middle of the PCR fragment?

c) What bands would be observed if the recombinant plasmid was cut with both EcoRI and HindIII at the same time?

8.3 If you started with 10 molecules of double stranded DNA template, what is the maximum number of molecules you would you have after 10 PCR cycles?

8.4 What is present in a PCR tube at the end of a successful amplification reaction? With this in mind, why do you usually only see a single, sharp band on a gel when it is analyzed by electrophoresis?

8.5 A coat protein from a particular virus can be used to immunize children against further infection. However, inoculation of children with proteins extracted from natural viruses sometimes causes fatal disease, due to contamination with live viruses. How could you use molecular biology to produce an optimal vaccine?

8.6 How would cloning be different if there were no selectable markers?

8.7 Research shows that a particular form of cancer is caused by a 200bp deletion in a particular human gene that is normally 2kb long. Only one mutant copy is needed to cause the disease.

a) Explain how you would use Southern blotting to diagnose the disease.

b) How would any of the blots appear if you hybridized and washed at very low temperature?

8.8 Refer to question 8.7.

a) Explain how you would detect the presence of the same deletion using PCR, rather than a Southern blot.

b) How would PCR products appear if you annealed at very low temperature?

8.9 You have a PCR fragment for a human olfactory receptor gene (perception of smells). You want to know what genes a dog might have that are related to this human gene. How can you use your PCR fragment and genomic DNA from a dog to find this out? Do you think dogs have more or less of these genes?

8.10 You add ligase to a reaction containing a sticky-ended plasmid and sticky-ended insert fragment, which both have compatible ends. Unbeknownst to you, someone in the lab left the stock of ligase enzyme out of the freezer overnight and it degraded (no longer works). Explain in detail what will happen in your ligation experiment in this situation should you try and transform with it.