# 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

exercise 1.2 solution b....isnt it lacking
I dnt get dis work well
what is one-to-one function
what is the procedure in solving quadratic equetion at least 6?
Almighty formula or by factorization...or by graphical analysis
Damian
I need to learn this trigonometry from A level.. can anyone help here?
yes am hia
Miiro
tanh2x =2tanhx/1+tanh^2x
cos(a+b)+cos(a-b)/sin(a+b)-sin(a-b)=cotb ... pls some one should help me with this..thanks in anticipation
f(x)=x/x+2 given g(x)=1+2x/1-x show that gf(x)=1+2x/3
proof
AUSTINE
sebd me some questions about anything ill solve for yall
cos(a+b)+cos(a-b)/sin(a+b)-sin(a-b)= cotb
favour
how to solve x²=2x+8 factorization?
x=2x+8 x-2x=2x+8-2x x-2x=8 -x=8 -x/-1=8/-1 x=-8 prove: if x=-8 -8=2(-8)+8 -8=-16+8 -8=-8 (PROVEN)
Manifoldee
x=2x+8
Manifoldee
×=2x-8 minus both sides by 2x
Manifoldee
so, x-2x=2x+8-2x
Manifoldee
then cancel out 2x and -2x, cuz 2x-2x is obviously zero
Manifoldee
so it would be like this: x-2x=8
Manifoldee
then we all know that beside the variable is a number (1): (1)x-2x=8
Manifoldee
so we will going to minus that 1-2=-1
Manifoldee
so it would be -x=8
Manifoldee
so next step is to cancel out negative number beside x so we get positive x
Manifoldee
so by doing it you need to divide both side by -1 so it would be like this: (-1x/-1)=(8/-1)
Manifoldee
so -1/-1=1
Manifoldee
so x=-8
Manifoldee
Manifoldee
so we should prove it
Manifoldee
x=2x+8 x-2x=8 -x=8 x=-8 by mantu from India
mantu
lol i just saw its x²
Manifoldee
x²=2x-8 x²-2x=8 -x²=8 x²=-8 square root(x²)=square root(-8) x=sq. root(-8)
Manifoldee
I mean x²=2x+8 by factorization method
Kristof
I think x=-2 or x=4
Kristof
x= 2x+8 ×=8-2x - 2x + x = 8 - x = 8 both sides divided - 1 -×/-1 = 8/-1 × = - 8 //// from somalia
Mohamed
hii
Amit
how are you
Dorbor
well
Biswajit
can u tell me concepts
Gaurav
Find the possible value of 8.5 using moivre's theorem
which of these functions is not uniformly cintinuous on (0, 1)? sinx
helo
Akash
hlo
Akash
Hello
Hudheifa
which of these functions is not uniformly continuous on 0,1
solve this equation by completing the square 3x-4x-7=0
X=7
Muustapha
=7
mantu
x=7
mantu
3x-4x-7=0 -x=7 x=-7
Kr
x=-7
mantu
9x-16x-49=0 -7x=49 -x=7 x=7
mantu
what's the formula
Modress
-x=7
Modress
new member
siame