# 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

By the definition, is such that 0!=1.why?
(1+cosA+IsinA)(1+cosB+isinB)/(cos@+isin@)(cos$+isin$)
hatdog
Mark
how we can draw three triangles of distinctly different shapes. All the angles will be cutt off each triangle and placed side by side with vertices touching
bsc F. y algebra and trigonometry pepper 2
given that x= 3/5 find sin 3x
4
DB
remove any signs and collect terms of -2(8a-3b-c)
-16a+6b+2c
Will
Joeval
(x2-2x+8)-4(x2-3x+5)
sorry
Miranda
x²-2x+9-4x²+12x-20 -3x²+10x+11
Miranda
x²-2x+9-4x²+12x-20 -3x²+10x+11
Miranda
(X2-2X+8)-4(X2-3X+5)=0 ?
master
The anwser is imaginary number if you want to know The anwser of the expression you must arrange The expression and use quadratic formula To find the answer
master
The anwser is imaginary number if you want to know The anwser of the expression you must arrange The expression and use quadratic formula To find the answer
master
Y
master
master
Soo sorry (5±Root11* i)/3
master
Mukhtar
2x²-6x+1=0
Ife
explain and give four example of hyperbolic function
What is the correct rational algebraic expression of the given "a fraction whose denominator is 10 more than the numerator y?
y/y+10
Mr
Find nth derivative of eax sin (bx + c).
Find area common to the parabola y2 = 4ax and x2 = 4ay.
Anurag
y2=4ax= y=4ax/2. y=2ax
akash
A rectangular garden is 25ft wide. if its area is 1125ft, what is the length of the garden
to find the length I divide the area by the wide wich means 1125ft/25ft=45
Miranda
thanks
Jhovie
What do you call a relation where each element in the domain is related to only one value in the range by some rules?
A banana.
Yaona
a function
Daniel
a function
emmanuel
given 4cot thither +3=0and 0°<thither <180° use a sketch to determine the value of the following a)cos thither
what are you up to?
nothing up todat yet
Miranda
hi
jai
hello
jai
Miranda Drice
jai
aap konsi country se ho
jai
which language is that
Miranda
I am living in india
jai
good
Miranda
what is the formula for calculating algebraic
I think the formula for calculating algebraic is the statement of the equality of two expression stimulate by a set of addition, multiplication, soustraction, division, raising to a power and extraction of Root. U believe by having those in the equation you will be in measure to calculate it
Miranda