Modeling Using Variation

Learning Objectives

In this section, you will:

  • Solve direct variation problems.
  • Solve inverse variation problems.
  • Solve problems involving joint variation.

A used-car company has just offered their best candidate, Nicole, a position in sales. The position offers 16% commission on her sales. Her earnings depend on the amount of her sales. For instance, if she sells a vehicle for $4,600, she will earn $736. She wants to evaluate the offer, but she is not sure how. In this section, we will look at relationships, such as this one, between earnings, sales, and commission rate.

Solving Direct Variation Problems

In the example above, Nicole’s earnings can be found by multiplying her sales by her commission. The formulae=0.16se=0.16stells us her earnings,e,e,come from the product of 0.16, her commission, and the sale price of the vehicle. If we create a table, we observe that as the sales price increases, the earnings increase as well, which should be intuitive. See (Figure).

ss, sales price e=0.16se=0.16s Interpretation
$4,600 e=0.16(4,600)=736e=0.16(4,600)=736 A sale of a $4,600 vehicle results in $736 earnings.
$9,200 e=0.16(9,200)=1,472e=0.16(9,200)=1,472 A sale of a $9,200 vehicle results in $1472 earnings.
$18,400 e=0.16(18,400)=2,944e=0.16(18,400)=2,944 A sale of a $18,400 vehicle results in $2944 earnings.

Notice that earnings are a multiple of sales. As sales increase, earnings increase in a predictable way. Double the sales of the vehicle from $4,600 to $9,200, and we double the earnings from $736 to $1,472. As the input increases, the output increases as a multiple of the input. A relationship in which one quantity is a constant multiplied by another quantity is called direct variation. Each variable in this type of relationship varies directly with the other.

(Figure) represents the data for Nicole’s potential earnings. We say that earnings vary directly with the sales price of the car. The formulay=kxny=kxnis used for direct variation. The valuekkis a nonzero constant greater than zero and is called the constant of variation. In this case,k=0.16k=0.16andn=1.n=1.We saw functions like this one when we discussed power functions.

Graph of y=(0.16)x where the horizontal axis is labeled, “s, Sales Price in Dollars”, and the vertical axis is labeled, “e, Earnings, $”.

Figure 1.

Direct Variation

Ifxandyxandyare related by an equation of the form

y=kxny=kxn

then we say that the relationship is direct variation and yy varies directly with, or is proportional to, thenthnthpower ofx.x.In direct variation relationships, there is a nonzero constant ratiok=yxn,k=yxn,wherekkis called the constant of variation, which help defines the relationship between the variables.

How To

Given a description of a direct variation problem, solve for an unknown.

  1. Identify the input,x,x,and the output,y.y.
  2. Determine the constant of variation. You may need to divideyyby the specified power ofxxto determine the constant of variation.
  3. Use the constant of variation to write an equation for the relationship.
  4. Substitute known values into the equation to find the unknown.

Solving a Direct Variation Problem

The quantityyyvaries directly with the cube ofx.x.Ify=25y=25whenx=2,x=2,findyywhenxxis 6.

Analysis

The graph of this equation is a simple cubic, as shown in (Figure).

Graph of y=25/8(x^3) with the labeled points (2, 25) and (6, 675).

Figure 2.

Do the graphs of all direct variation equations look like (Figure)?

No. Direct variation equations are power functions—they may be linear, quadratic, cubic, quartic, radical, etc. But all of the graphs pass through(0,0).(0,0).

Try It

The quantityyyvaries directly with the square ofx.x.Ify=24y=24whenx=3,x=3,findyywhenxxis 4.

Solving Inverse Variation Problems

Water temperature in an ocean varies inversely to the water’s depth. The formulaT=14,000dT=14,000dgives us the temperature in degrees Fahrenheit at a depth in feet below Earth’s surface. Consider the Atlantic Ocean, which covers 22% of Earth’s surface. At a certain location, at the depth of 500 feet, the temperature may be 28°F.

If we create (Figure), we observe that, as the depth increases, the water temperature decreases.

d,d,depth T=14,000dT=14,000d Interpretation
500 ft 14,000500=2814,000500=28 At a depth of 500 ft, the water temperature is 28° F.
1000 ft 14,0001000=1414,0001000=14 At a depth of 1,000 ft, the water temperature is 14° F.
2000 ft 14,0002000=714,0002000=7 At a depth of 2,000 ft, the water temperature is 7° F.

We notice in the relationship between these variables that, as one quantity increases, the other decreases. The two quantities are said to be inversely proportional and each term varies inversely with the other. Inversely proportional relationships are also called inverse variations.

For our example, (Figure) depicts the inverse variation. We say the water temperature varies inversely with the depth of the water because, as the depth increases, the temperature decreases. The formulay=kxy=kxfor inverse variation in this case usesk=14,000.k=14,000.

Graph of y=(14000)/x where the horizontal axis is labeled, “Depth, d (ft)”, and the vertical axis is labeled, “Temperature, T (Degrees Fahrenheit)”.

Figure 3.

Inverse Variation

Ifxxandyyare related by an equation of the form

y=kxny=kxn

wherekkis a nonzero constant, then we say thatyy varies inversely with thenthnthpower ofx.x.In inversely proportional relationships, or inverse variations, there is a constant multiplek=xny.k=xny.

Writing a Formula for an Inversely Proportional Relationship

A tourist plans to drive 100 miles. Find a formula for the time the trip will take as a function of the speed the tourist drives.

How To

Given a description of an indirect variation problem, solve for an unknown.

  1. Identify the input,x,x,and the output,y.y.
  2. Determine the constant of variation. You may need to multiplyyyby the specified power ofxxto determine the constant of variation.
  3. Use the constant of variation to write an equation for the relationship.
  4. Substitute known values into the equation to find the unknown.

Solving an Inverse Variation Problem

A quantityyyvaries inversely with the cube ofx.x.Ify=25y=25whenx=2,x=2,findyywhenxxis 6.

Analysis

The graph of this equation is a rational function, as shown in (Figure).

Graph of y=25/(x^3) with the labeled points (2, 25) and (6, 25/27).

Figure 4.

Try It

A quantityyyvaries inversely with the square ofx.x.Ify=8y=8whenx=3,x=3,findyywhenxxis 4.

Solving Problems Involving Joint Variation

Many situations are more complicated than a basic direct variation or inverse variation model. One variable often depends on multiple other variables. When a variable is dependent on the product or quotient of two or more variables, this is called joint variation. For example, the cost of busing students for each school trip varies with the number of students attending and the distance from the school. The variablec,c,cost, varies jointly with the number of students,n,n,and the distance,d.d.

Joint Variation

Joint variation occurs when a variable varies directly or inversely with multiple variables.

For instance, ifxxvaries directly with bothyyandz,z, we havex=kyz.x=kyz.Ifxxvaries directly withyyand inversely withz,z,we havex=kyz.x=kyz.Notice that we only use one constant in a joint variation equation.

Solving Problems Involving Joint Variation

A quantityxxvaries directly with the square ofyyand inversely with the cube root ofz.z.Ifx=6x=6wheny=2y=2 andz=8,z=8,findxxwheny=1y=1 and z=27.z=27.

Try It

A quantityxxvaries directly with the square ofyyand inversely withz.z.Ifx=40x=40wheny=4y=4andz=2,z=2,findxxwheny=10y=10
andz=25.z=25.

Access these online resources for additional instruction and practice with direct and inverse variation.

Visit this website for additional practice questions from Learningpod.

Key Equations

Direct variation y=kxn,k is a nonzero constant.y=kxn,k is a nonzero constant.
Inverse variation y=kxn,k is a nonzero constant.y=kxn,k is a nonzero constant.

Key Concepts

  • A relationship where one quantity is a constant multiplied by another quantity is called direct variation. See (Figure).
  • Two variables that are directly proportional to one another will have a constant ratio.
  • A relationship where one quantity is a constant divided by another quantity is called inverse variation. See (Figure).
  • Two variables that are inversely proportional to one another will have a constant multiple. See (Figure).
  • In many problems, a variable varies directly or inversely with multiple variables. We call this type of relationship joint variation. See (Figure).

Section Exercises

Verbal

What is true of the appearance of graphs that reflect a direct variation between two variables?

If two variables vary inversely, what will an equation representing their relationship look like?

Is there a limit to the number of variables that can vary jointly? Explain.

Algebraic

For the following exercises, write an equation describing the relationship of the given variables.

yyvaries directly asxxand whenx=6,y=12.x=6,y=12.

yyvaries directly as the square ofxxand whenx=4,y=80.x=4,y=80.

yyvaries directly as the square root ofxxand whenx=36,y=24.x=36,y=24.

yyvaries directly as the cube ofxxand whenx=36,y=24.x=36,y=24.

yyvaries directly as the cube root ofxxand whenx=27,y=15.x=27,y=15.

yyvaries directly as the fourth power ofxxand whenx=1,y=6.x=1,y=6.

yyvaries inversely asxxand whenx=4,y=2.x=4,y=2.

yyvaries inversely as the square ofxxand whenx=3,y=2.x=3,y=2.

yyvaries inversely as the cube ofxxand whenx=2,y=5.x=2,y=5.

yyvaries inversely as the fourth power ofxxand whenx=3,y=1.x=3,y=1.

yyvaries inversely as the square root ofxxand whenx=25,y=3.x=25,y=3.

yyvaries inversely as the cube root ofxxand whenx=64,y=5.x=64,y=5.

yyvaries jointly withxxandzzand whenx=2x=2andz=3,y=36.z=3,y=36.

yyvaries jointly asx,z,x,z,andwwand whenx=1,z=2,w=5,x=1,z=2,w=5,theny=100.y=100.

yyvaries jointly as the square ofxxand the square ofzzand whenx=3x=3andz=4,z=4,theny=72.y=72.

yyvaries jointly asxxand the square root ofzzand whenx=2x=2andz=25,z=25,theny=100.y=100.

yyvaries jointly as the square ofxxthe cube ofzzand the square root ofW.W.Whenx=1,z=2,x=1,z=2,andw=36,w=36,theny=48.y=48.

yy
varies jointly asxandzxandzand inversely asw.w.Whenx=3,z=5,x=3,z=5,andw=6,w=6,theny=10.y=10.

yyvaries jointly as the square ofxxand the square root ofzzand inversely as the cube ofw.\hspace{0.17em}w.\hspace{0.17em}Whenx=3,z=4,x=3,z=4,andw=3,w=3,theny=6.y=6.

yyvaries jointly asxxandzzand inversely as the square root ofwwand the square oft\hspace{0.17em}.t\hspace{0.17em}.Whenx=3,z=1,w=25,x=3,z=1,w=25,andt=2,t=2,theny=6.y=6.

Numeric

For the following exercises, use the given information to find the unknown value.

yyvaries directly asx.x.Whenx=3,x=3,theny=12.y=12.Findyywnehx=20.x=20.

yyvaries directly as the square ofx.x.Whenx=2,x=2,theny=16.y=16.Findyywhenx=8.x=8.

yyvaries directly as the cube ofx.x.Whenx=3,x=3,theny=5.y=5.Findyywhenx=4.x=4.

yyvaries directly as the square root ofx.x.Whenx=16,x=16,theny=4.y=4.Findyywhenx=36.x=36.

yyvaries directly as the cube root ofx.x.Whenx=125,x=125,theny=15.y=15.Findyywhenx=1,000.x=1,000.

yyvaries inversely withx.x.Whenx=3,x=3,theny=2.y=2.Findyywhenx=1.x=1.

yyvaries inversely with the square ofx.x.Whenx=4,x=4,theny=3.y=3.Findyywhenx=2.x=2.

yyvaries inversely with the cube ofx.x.Whenx=3,x=3,theny=1.y=1.Findyywhenx=1.x=1.

yyvaries inversely with the square root ofx.x.Whenx=64,x=64,theny=12.y=12.Findyywhenx=36.x=36.

yyvaries inversely with the cube root ofx.x.Whenx=27,x=27,theny=5.y=5.Findyywhenx=125.x=125.

yyvaries jointly asxandz.xandz.Whenx=4x=4andz=2,z=2,theny=16.y=16.Findyywhenx=3x=3andz=3.z=3.

yyvaries jointly asx,z,andw.x,z,andw.Whenx=2,x=2,z=1,z=1,andw=12,w=12,theny=72.y=72.Findyywhenx=1,x=1,z=2,z=2, andw=3.w=3.

yyvaries jointly asxxand the square ofz.z.Whenx=2x=2andz=4,z=4,theny=144.y=144.Findyywhenx=4x=4andz=5.z=5.

yyvaries jointly as the square ofxxand the square root ofz.z.Whenx=2x=2andz=9,z=9,theny=24.y=24.Findyywhenx=3x=3andz=25.z=25.

yyvaries jointly asxxandzzand inversely asw.w.Whenx=5,x=5,z=2,z=2,andw=20,w=20,theny=4.y=4.Findyywhenx=3x=3andz=8,z=8,andw=48.w=48.

yyvaries jointly as the square ofxxand the cube ofzzand inversely as the square root ofw\.w\.Whenx=2,x=2,z=2,z=2,andw=64,w=64,theny=12.y=12.Findyywhenx=1,x=1,z=3,z=3,andw=4.w=4.

yyvaries jointly as the square ofxxand ofzzand inversely as the square root ofwwand oft\.t\.Whenx=2,x=2,z=3,z=3,w=16,w=16,andt=3,t=3,theny=1.y=1.Findyywhenx=3,x=3,z=2,z=2,w=36,w=36,andt=5.t=5.

Technology

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

yyvaries directly with the square ofxxand whenx=2,y=3.x=2,y=3.

yyvaries directly as the cube ofxxand whenx=2,y=4.x=2,y=4.

yyvaries directly as the square root ofxxand whenx=36,y=2.x=36,y=2.

yyvaries inversely withxxand whenx=6,y=2.x=6,y=2.

yyvaries inversely as the square ofxxand whenx=1,y=4.x=1,y=4.

Extensions

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

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.

Using Earth’s distance of 150 million kilometers, find the equation relatingTTanda.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.

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 distancessthat an object falls varies directly with the square of the time,t,t,of the fall. If an object falls 16 feet in one second, how long for it to fall 144 feet?

The velocityvvof a falling object varies directly to the time,t,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?

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?

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?

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.

The force exerted by the wind on a plane surface varies jointly with the square of the velocity of the wind and with the area of the plane surface. If the area of the surface is 40 square feet surface and the wind velocity is 20 miles per hour, the resulting force is 15 pounds. Find the force on a surface of 65 square feet with a velocity of 30 miles per hour.

The horsepower (hp) that a shaft can safely transmit varies jointly with its speed (in revolutions per minute (rpm) and the cube of the diameter. If the shaft of a certain material 3 inches in diameter can transmit 45 hp at 100 rpm, what must the diameter be in order to transmit 60 hp at 150 rpm?

The kinetic energyKKof a moving object varies jointly with its massmmand the square of its velocityv.v.If an object weighing 40 kilograms with a velocity of 15 meters per second has a kinetic energy of 1000 joules, find the kinetic energy if the velocity is increased to 20 meters per second.

Chapter Review Exercises

Quadratic Functions

For the following exercises, write the quadratic function in standard form. Then give the vertex and axes intercepts. Finally, graph the function.

f(x)=x24x5f(x)=x24x5

f(x)=2x24xf(x)=2x24x

For the following exercises, find the equation of the quadratic function using the given information.

The vertex is(2,3)(2,3)and a point on the graph is(3,6).(3,6).

The vertex is(3,6.5)(3,6.5)and a point on the graph is(2,6).(2,6).

For the following exercises, complete the task.

A rectangular plot of land is to be enclosed by fencing. One side is along a river and so needs no fence. If the total fencing available is 600 meters, find the dimensions of the plot to have maximum area.

An object projected from the ground at a 45 degree angle with initial velocity of 120 feet per second has height,h,h,in terms of horizontal distance traveled,x,x,given byh(x)=32(120)2x2+x.h(x)=32(120)2x2+x.Find the maximum height the object attains.

Power Functions and Polynomial Functions

For the following exercises, determine if the function is a polynomial function and, if so, give the degree and leading coefficient.

f(x)=4x53x3+2x1f(x)=4x53x3+2x1

f(x)=5x+1x2f(x)=5x+1x2

f(x)=x2(36x+x2)f(x)=x2(36x+x2)

For the following exercises, determine end behavior of the polynomial function.

f(x)=2x4+3x35x2+7f(x)=2x4+3x35x2+7

f(x)=4x36x2+2f(x)=4x36x2+2

f(x)=2x2(1+3xx2)

Graphs of Polynomial Functions

For the following exercises, find all zeros of the polynomial function, noting multiplicities.

f(x)=(x+3)2(2x1)(x+1)3

f(x)=x5+4x4+4x3

f(x)=x34x2+x4

For the following exercises, based on the given graph, determine the zeros of the function and note multiplicity.

Graph of an odd-degree polynomial with two turning points.
Graph of an even-degree polynomial with two turning points.

Use the Intermediate Value Theorem to show that at least one zero lies between 2 and 3 for the functionf(x)=x35x+1

Dividing Polynomials

For the following exercises, use long division to find the quotient and remainder.

x32x2+4x+4x2

3x44x2+4x+8x+1

For the following exercises, use synthetic division to find the quotient. If the divisor is a factor, then write the factored form.

x32x2+5x1x+3

x3+4x+10x3

2x3+6x211x12x+4

3x4+3x3+2x+2x+1

Zeros of Polynomial Functions

For the following exercises, use the Rational Zero Theorem to help you solve the polynomial equation.

2x33x218x8=0

3x3+11x2+8x4=0

2x417x3+46x243x+12=0

4x4+8x3+19x2+32x+12=0

For the following exercises, use Descartes’ Rule of Signs to find the possible number of positive and negative solutions.

x33x22x+4=0

2x4x3+4x25x+1=0

Rational Functions

For the following exercises, find the intercepts and the vertical and horizontal asymptotes, and then use them to sketch a graph of the function.

f(x)=x+2x5

f(x)=x2+1x24

f(x)=3x227x2+x2

f(x)=x+2x29

For the following exercises, find the slant asymptote.

f(x)=x21x+2

f(x)=2x3x2+4x2+1

Inverses and Radical Functions

For the following exercises, find the inverse of the function with the domain given.

f(x)=(x2)2,x2

f(x)=(x+4)23,x4

f(x)=x2+6x2,x3

f(x)=2x33

f(x)=4x+53

f(x)=x32x+1

Modeling Using Variation

For the following exercises, find the unknown value.

y varies directly as the square ofx. If whenx=3, y=36,findyifx=4.

y varies inversely as the square root ofx If whenx=25, y=2,findyifx=4.

yvaries jointly as the cube ofxand asz.If whenx=1andz=2,y=6,findyifx=2andz=3.

yvaries jointly asxand the square ofzand inversely as the cube ofw.If whenx=3,z=4,andw=2,y=48,findyifx=4,z=5,andw=3.

For the following exercises, solve the application problem.

The weight of an object above the surface of the earth varies inversely with the distance from the center of the earth. If a person weighs 150 pounds when he is on the surface of the earth (3,960 miles from center), find the weight of the person if he is 20 miles above the surface.

The volumeVof an ideal gas varies directly with the temperatureTand inversely with the pressure P. A cylinder contains oxygen at a temperature of 310 degrees K and a pressure of 18 atmospheres in a volume of 120 liters. Find the pressure if the volume is decreased to 100 liters and the temperature is increased to 320 degrees K.

Chapter Test

Give the degree and leading coefficient of the following polynomial function.

f(x)=x3(36x22x2)

Determine the end behavior of the polynomial function.

f(x)=8x33x2+2x4

f(x)=2x2(43x5x2)

Write the quadratic function in standard form. Determine the vertex and axes intercepts and graph the function.

f(x)=x2+2x8

Given information about the graph of a quadratic function, find its equation.

Vertex(2,0)and point on graph(4,12).

Solve the following application problem.

A rectangular field is to be enclosed by fencing. In addition to the enclosing fence, another fence is to divide the field into two parts, running parallel to two sides. If 1,200 feet of fencing is available, find the maximum area that can be enclosed.

Find all zeros of the following polynomial functions, noting multiplicities.

f(x)=(x3)3(3x1)(x1)2

f(x)=2x612x5+18x4

Based on the graph, determine the zeros of the function and multiplicities.

Graph of an odd-degree polynomial with two turning points.

Use long division to find the quotient.

2x3+3x4x+2

Use synthetic division to find the quotient. If the divisor is a factor, write the factored form.

x4+3x24x2

2x3+5x27x12x+3

Use the Rational Zero Theorem to help you find the zeros of the polynomial functions.

f(x)=2x3+5x26x9

f(x)=4x4+8x3+21x2+17x+4

f(x)=4x4+16x3+13x215x18

f(x)=x5+6x4+13x3+14x2+12x+8

Given the following information about a polynomial function, find the function.

It has a double zero atx=3and zeros atx=1andx=2. Its y-intercept is(0,12).

It has a zero of multiplicity 3 atx=12and another zero atx=3. It contains the point(1,8).

Use Descartes’ Rule of Signs to determine the possible number of positive and negative solutions.

8x321x2+6=0

For the following rational functions, find the intercepts and horizontal and vertical asymptotes, and sketch a graph.

f(x)=x+4x22x3

f(x)=x2+2x3x24

Find the slant asymptote of the rational function.

f(x)=x2+3x3x1

Find the inverse of the function.

f(x)=x2+4

f(x)=3x34

f(x)=2x+33x1

Find the unknown value.

yvaries inversely as the square ofxand whenx=3,y=2.Findyifx=1.

yvaries jointly withxand the cube root ofz.If whenx=2andz=27,y=12,findyifx=5andz=8.

Solve the following application problem.

The distance a body falls varies directly as the square of the time it falls. If an object falls 64 feet in 2 seconds, how long will it take to fall 256 feet?

Glossary

constant of variation
the non-zero valuek
that helps define the relationship between variables in direct or inverse variation
direct variation
the relationship between two variables that are a constant multiple of each other; as one quantity increases, so does the other
inverse variation
the relationship between two variables in which the product of the variables is a constant
inversely proportional
a relationship where one quantity is a constant divided by the other quantity; as one quantity increases, the other decreases
joint variation
a relationship where a variable varies directly or inversely with multiple variables
varies directly
a relationship where one quantity is a constant multiplied by the other quantity
varies inversely
a relationship where one quantity is a constant divided by the other quantity