# 5.8 Modeling using variation  (Page 5/14)

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$\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ varies jointly as the square of $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ and of $\text{\hspace{0.17em}}z\text{\hspace{0.17em}}$ and inversely as the square root of $\text{\hspace{0.17em}}w\text{\hspace{0.17em}}$ and of $\text{\hspace{0.17em}}t\text{\hspace{0.17em}.}$ When $\text{\hspace{0.17em}}x=2,\text{\hspace{0.17em}}$ $z=3,\text{\hspace{0.17em}}$ $w=16,\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}t=3,\text{\hspace{0.17em}}$ then $\text{\hspace{0.17em}}y=1.\text{\hspace{0.17em}}$ Find $\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ when $\text{\hspace{0.17em}}x=3,\text{\hspace{0.17em}}$ $z=2,\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}t=5.$

## Technology

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

$\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ varies directly with the square of $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ and when

$y=\frac{3}{4}{x}^{2}$

$\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ varies directly as the cube of $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ and when

$\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ varies directly as the square root of $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ and when

$y=\frac{1}{3}\sqrt{x}$

$\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ varies inversely with $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ and when

$\text{\hspace{0.17em}}y\text{\hspace{0.17em}}$ varies inversely as the square of $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ and when

$y=\frac{4}{{x}^{2}}$

## Extensions

For the following exercises, use Kepler’s Law, which states that the square of the time, $\text{\hspace{0.17em}}T,\text{\hspace{0.17em}}$ required for a planet to orbit the Sun varies directly with the cube of the mean distance, $\text{\hspace{0.17em}}a,\text{\hspace{0.17em}}$ 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 $\text{\hspace{0.17em}}T\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}a.\text{\hspace{0.17em}}$

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

Using Earth’s distance of 150 million kilometers, find the equation relating $\text{\hspace{0.17em}}T\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}a.$

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

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.

## Real-world applications

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

The distance $\text{\hspace{0.17em}}s\text{\hspace{0.17em}}$ that an object falls varies directly with the square of the time, $\text{\hspace{0.17em}}t,\text{\hspace{0.17em}}$ of the fall. If an object falls 16 feet in one se c ond, how long for it to fall 144 feet?

3 seconds

The velocity $\text{\hspace{0.17em}}v\text{\hspace{0.17em}}$ of a falling object varies directly to the time, $\text{\hspace{0.17em}}t,\text{\hspace{0.17em}}$ of the fall. If after 2 seconds, the velocity of the object is 64 feet per second, what is the velocity after 5 seconds?

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

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?

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

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.

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

answer and questions in exercise 11.2 sums
how do u calculate inequality of irrational number?
Alaba
give me an example
Chris
and I will walk you through it
Chris
cos (-z)= cos z .
what is a algebra
(x+x)3=?
what is the identity of 1-cos²5x equal to?
__john __05
Kishu
Hi
Abdel
hi
Ye
hi
Nokwanda
C'est comment
Abdel
Hi
Amanda
hello
SORIE
Hiiii
Chinni
hello
Ranjay
hi
ANSHU
hiiii
Chinni
h r u friends
Chinni
yes
Hassan
so is their any Genius in mathematics here let chat guys and get to know each other's
SORIE
I speak French
Abdel
okay no problem since we gather here and get to know each other
SORIE
hi im stupid at math and just wanna join here
Yaona
lol nahhh none of us here are stupid it's just that we have Fast, Medium, and slow learner bro but we all going to work things out together
SORIE
it's 12
what is the function of sine with respect of cosine , graphically
tangent bruh
Steve
cosx.cos2x.cos4x.cos8x
sinx sin2x is linearly dependent
what is a reciprocal
The reciprocal of a number is 1 divided by a number. eg the reciprocal of 10 is 1/10 which is 0.1
Shemmy
Reciprocal is a pair of numbers that, when multiplied together, equal to 1. Example; the reciprocal of 3 is ⅓, because 3 multiplied by ⅓ is equal to 1
Jeza
each term in a sequence below is five times the previous term what is the eighth term in the sequence
I don't understand how radicals works pls
How look for the general solution of a trig function
stock therom F=(x2+y2) i-2xy J jaha x=a y=o y=b
sinx sin2x is linearly dependent
cr
root under 3-root under 2 by 5 y square
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
cosA\1+sinA=secA-tanA
Wrong question
why two x + seven is equal to nineteen.
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