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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

root under 3-root under 2 by 5 y square
Himanshu Reply
The sum of the first n terms of a certain series is 2^n-1, Show that , this series is Geometric and Find the formula of the n^th
amani Reply
cosA\1+sinA=secA-tanA
Aasik Reply
why two x + seven is equal to nineteen.
Kingsley Reply
The numbers cannot be combined with the x
Othman
2x + 7 =19
humberto
2x +7=19. 2x=19 - 7 2x=12 x=6
Yvonne
because x is 6
SAIDI
what is the best practice that will address the issue on this topic? anyone who can help me. i'm working on my action research.
Melanie Reply
simplify each radical by removing as many factors as possible (a) √75
Jason Reply
how is infinity bidder from undefined?
Karl Reply
what is the value of x in 4x-2+3
Vishal Reply
give the complete question
Shanky
4x=3-2 4x=1 x=1+4 x=5 5x
Olaiya
hi can you give another equation I'd like to solve it
Daniel
what is the value of x in 4x-2+3
Olaiya
if 4x-2+3 = 0 then 4x = 2-3 4x = -1 x = -(1÷4) is the answer.
Jacob
4x-2+3 4x=-3+2 4×=-1 4×/4=-1/4
LUTHO
then x=-1/4
LUTHO
4x-2+3 4x=-3+2 4x=-1 4x÷4=-1÷4 x=-1÷4
LUTHO
A research student is working with a culture of bacteria that doubles in size every twenty minutes. The initial population count was  1350  bacteria. Rounding to five significant digits, write an exponential equation representing this situation. To the nearest whole number, what is the population size after  3  hours?
David Reply
v=lbh calculate the volume if i.l=5cm, b=2cm ,h=3cm
Haidar Reply
Need help with math
Peya
can you help me on this topic of Geometry if l help you
litshani
( cosec Q _ cot Q ) whole spuare = 1_cosQ / 1+cosQ
Aarav Reply
A guy wire for a suspension bridge runs from the ground diagonally to the top of the closest pylon to make a triangle. We can use the Pythagorean Theorem to find the length of guy wire needed. The square of the distance between the wire on the ground and the pylon on the ground is 90,000 feet. The square of the height of the pylon is 160,000 feet. So, the length of the guy wire can be found by evaluating √(90000+160000). What is the length of the guy wire?
Maxwell Reply
the indicated sum of a sequence is known as
Arku Reply
how do I attempted a trig number as a starter
Tumwe Reply
cos 18 ____ sin 72 evaluate
Het Reply
Practice Key Terms 7

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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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