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y varies jointly as the square of x and of z and inversely as the square root of w and of t  . When x = 2 , z = 3 , w = 16 , and t = 3 , then y = 1. Find y when x = 3 , z = 2 ,   w = 36 , and t = 5.

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Technology

For the following exercises, use a calculator to graph the equation implied by the given variation.

y varies directly with the square of x and when x = 2 ,   y = 3.

y = 3 4 x 2

Graph of y=3/4(x^2).
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y varies directly as the cube of x and when x = 2 ,   y = 4.

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y varies directly as the square root of x and when x = 36 ,   y = 2.

y = 1 3 x

Graph of y=1/3sqrt(x).
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y varies inversely with x and when x = 6 ,   y = 2.

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y varies inversely as the square of x and when x = 1 ,   y = 4.

y = 4 x 2

Graph of y=4/(x^2).
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Extensions

For the following exercises, use Kepler’s Law, which states that the square of the time, T , required for a planet to orbit the Sun varies directly with the cube of the mean distance, a , that the planet is from the Sun.

Using Earth’s time of 1 year and mean distance of 93 million miles, find the equation relating T and a .

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Use the result from the previous exercise to determine the time required for Mars to orbit the Sun if its mean distance is 142 million miles.

1.89 years

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Using Earth’s distance of 150 million kilometers, find the equation relating T and a .

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Use the result from the previous exercise to determine the time required for Venus to orbit the Sun if its mean distance is 108 million kilometers.

0.61 years

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Using Earth’s distance of 1 astronomical unit (A.U.), determine the time for Saturn to orbit the Sun if its mean distance is 9.54 A.U.

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Real-world applications

For the following exercises, use the given information to answer the questions.

The distance s that an object falls varies directly with the square of the time, t , of the fall. If an object falls 16 feet in one se c ond, how long for it to fall 144 feet?

3 seconds

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The velocity v of a falling object varies directly to the time, t , of the fall. If after 2 seconds, the velocity of the object is 64 feet per second, what is the velocity after 5 seconds?

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The rate of vibration of a string under constant tension varies inversely with the length of the string. If a string is 24 inches long and vibrates 128 times per second, what is the length of a string that vibrates 64 times per second?

48 inches

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The volume of a gas held at constant temperature varies indirectly as the pressure of the gas. If the volume of a gas is 1200 cubic centimeters when the pressure is 200 millimeters of mercury, what is the volume when the pressure is 300 millimeters of mercury?

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The weight of an object above the surface of Earth varies inversely with the square of the distance from the center of Earth. If a body weighs 50 pounds when it is 3960 miles from Earth’s center, what would it weigh it were 3970 miles from Earth’s center?

49.75 pounds

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The intensity of light measured in foot-candles varies inversely with the square of the distance from the light source. Suppose the intensity of a light bulb is 0.08 foot-candles at a distance of 3 meters. Find the intensity level at 8 meters.

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The current in a circuit varies inversely with its resistance measured in ohms. When the current in a circuit is 40 amperes, the resistance is 10 ohms. Find the current if the resistance is 12 ohms.

33.33 amperes

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Questions & Answers

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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