1.8e: Exercises - Variation

A: Translate Words into a Formula

Exercise $\PageIndex{A}$ 

$\bigstar$ 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.

Answers to odd exercises:

1. $D=kt$

3. $d=kv^{2}$

5. $V = \frac{k}{p}$

7. $I=krt$

B: Translate Words and Find a Formula

Exercise $\PageIndex{B}$ 

$\bigstar$  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 root of $w$, where $y=14$ when $x=4, w=9$ and $z=2$.

Answers to odd exercises: 

9. $y=5x$ 11. $y=4x$

13. $y=\frac{1}{3}x$ 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. $y = \frac { 12 \sqrt { x } } { z ^ { 2 } }$

C: Direct variation problems

Exercise $\PageIndex{C}$: Direct variation problems

$\bigstar$  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?

$\bigstar$ 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?

$\bigstar$ 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 to odd exercises:

29. $$382.50$ 31. $$26.65$ 33. $2π$

35. $36π$ square meters 37. $50$ cubic centimeters 39. $\frac{1}{4}$ inches/pound

41. $7.5$ inches 43. $49$ feet 45. $1.4$ seconds

D: Inverse Variation Problems

Exercise $\PageIndex{D}$

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. The amount of illumination $I$ is inversely proportional to the square of the distance $d$ from a light source. 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?

$\bigstar$  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 to odd exercises:

47. $4$ feet

49. $1,120$ foot-candles

51. $108$ cubic inches

E: Joint and Combined Variation

Exercise $\PageIndex{E}$ 

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.

$\bigstar$  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 to odd exercises:

53. $4$ men 55. $6.7 \times 10 ^ { - 11 } \mathrm { Nm } ^ { 2 } / \mathrm { kg } ^ { 2 }$

57. $1.30 \times 10 ^ { 15 } \mathrm { N }$ 59. $y$ changes by a factor of $4$

61. $y$ remains unchanged

F: More Variation Problems

Exercise $\PageIndex{F}$ 

$\bigstar$ 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 113.1 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 weighs 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? 

Answers to odd exercises:

71. 432 calories	73. 78 pounds	75. 254.5 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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