College AlgebraSixth EditionJames Stewart Lothar Redlin Saleem Watson

Polynomial and

Rational Functions3

Modeling Variation3.8

Fundamentals

When scientists talk about a mathematical

model for a real-world phenomenon, they

often mean an equation that describes

the dependence of one physical quantity on

another.

• For instance, the model may describe the population of an animal species as a function of

time or the pressure of a gas as a function of its volume.

Fundamentals

In this section, we study a kind of

modeling called variation.

Direct Variation

Direct Variation

One type of variation is called direct

variation; it occurs when one

quantity is a constant multiple of the

other.

We use a function of the form

f(x) = kx to model this dependence.

Direct Variation

If the quantities x and y are related by

an equation y = kx for some constant k ≠ 0,

we say that:• y varies directly as x.

• y is directly proportional to x.

• y is proportional to x.

The constant k is called the constant

of proportionality.

Direct Variation

Recall that the graph of an equation

of the form

y = mx + b

is a line with:• Slope m• y-intercept b

Direct Variation

So, the graph of an equation y = kx

that describes direct variation is

a line with:• Slope k• y-intercept 0

E.g. 1—Direct Variation

During a thunderstorm, you see the lightning

before you hear the thunder because light

travels much faster than sound.

• The distance between you and the storm varies directly as the time interval between the lightning and the thunder.

E.g. 1—Direct Variation

(a)Suppose that the thunder from

a storm 5,400 ft away takes 5 s

to reach you.

• Determine the constant of proportionality and write the equation for the variation.

E.g. 1—Direct Variation

(b) Sketch the graph of this equation.

• What does the constant of proportionality represent?

(c) If the time interval between the lightning

and thunder is now 8 s, how far away is

the storm?

E.g. 1—Direct Variation

Let d be the distance from you to the storm

and let t be the length of the time interval.

• We are given that d varies directly as t.

• So, d = kt

where k is a constant.

Example (a)

E.g. 1—Direct Variation

To find k, we use the fact that t = 5

when d = 5400.

• Substituting these values in the equation, we get: 5400 = k(5)

Example (a)

54001080

5k

E.g. 1—Direct Variation

Substituting this value of k in the equation

for d, we obtain:

d = 1080t

as the equation for d as a function of t.

Example (a)

E.g. 1—Direct Variation

The graph of the equation d = 1080t is

a line through the origin with slope 1080.

• The constant k = 1080 is the approximate speed of sound (in ft/s).

Example (b)

E.g. 1—Direct Variation

When t = 8, we have:

d = 1080 ∙ 8 = 8640

• So, the storm is 8640 ft ≈ 1.6 mi away.

Example (c)

Inverse Variation

Inverse Variation

Another function that is frequently used

in mathematical modeling is

where k is a constant.

( )k

f xx

Inverse Variation

If the quantities x and y are related by

the equation

for some constant k ≠ 0,

we say that:• y is inversely proportional to x.

• y varies inversely as x.

ky

x

Inverse Variation

The graph of y = k/x for x > 0 is shown

for the case k > 0.

• It gives a picture of what happens when y is inversely proportional to x.

E.g. 2—Inverse Variation

Boyle’s Law states that:

• When a sample of gas is compressed at a constant temperature, the pressure of the gas is inversely proportional to the volume of the gas.

E.g. 2—Inverse Variation

(a) Suppose the pressure of a sample

of air that occupies 0.106 m3 at 25°C

is 50 kPa.

• Find the constant of proportionality.

• Write the equation that expresses the inverse proportionality.

E.g. 2—Inverse Variation

(b)If the sample expands to

a volume of 0.3 m3, find

the new pressure.

E.g. 2—Inverse Variation

Let P be the pressure of the sample of gas

and let V be its volume.

• Then, by the definition of inverse proportionality, we have:

where k is a constant.

Example (a)

kP

V

E.g. 2—Inverse Variation

To find k, we use the fact that P = 50

when V = 0.106.

• Substituting these values in the equation, we get:

k = (50)(0.106) = 5.3

500.106

k

Example (a)

E.g. 2—Inverse Variation

Putting this value of k in the equation

for P, we have:

Example (a)

5.3P

V

E.g. 2—Inverse Variation

When V = 0.3, we have:

• So, the new pressure is about 17.7 kPa.

Example (b)

5.317.7

0.3P

Combining Different Types of

Variation

Combining Different Types of Variation

In the sciences, relationships between three

or more variables are common, and any

combination of the different types of

proportionality that we have discussed is

possible.

Combining Different Types of Variation

If the quantities x, y, and z are related by

the equation

z = kxy

where k is a nonzero constant,

we say that:• z varies jointly as x and y.

• z is jointly proportional to x and y.

Combining Different Types of Variation

In the sciences, relationships between

three or more variables are common.

• Any combination of the different types of proportionality that we have discussed is possible.

• For example, if

we say that z is proportional to x and inversely proportional to y.

xz k

y

E.g. 4—Newton’s Law of Gravitation

Newton’s Law of Gravitation says that:

Two objects with masses m1 and m2 attract

each other with a force F that is jointly

proportional to their masses and inversely

proportional to the square of the distance r

between the objects.

• Express the law as an equation.

E.g. 4—Newton’s Law of Gravitation

Using the definitions of joint and inverse

variation, and the traditional notation G for

the gravitational constant of proportionality,

we have:

1 22

m mF G

r

Gravitational Force

If m1 and m2 are fixed masses,

then the gravitational force between them

is:

F = C/r2

where C = Gm1m2 is a constant.

Gravitational Force

The figure shows the graph of this

equation for r > 0 with C = 1.

• Observe how the gravitational attraction decreases with increasing distance.