Intermediate Algebra (11th edition)

Solving an Applied Problem Using a Quadratic Function

If an object is projected upward from the top of a 144-ft building at 112 ft per sec, its

position (in feet above the ground) is given by

where tis time in seconds after it was projected. When does it hit the ground?

When the object hits the ground, its distance above the ground is 0. We must find

the value of tthat makes

Let

Divide by 16.

Substitute into the quadratic formula.

Use a calculator.

The solutions are or Time cannot be negative, so we discard the neg-

ative solution. The object hits the ground about 8.1 sec after it is projected.

tL8.1 tL-1.1.

t=

7  285

2

L

7 9.2

2

t=

- 1 - 72  21 - 722 - 41121 - 92

2112

0 =t^2 - 7 t- 9 -

0 =- 16 t^2 + 112 t+ 144 s 1 t 2 =0.

s 1 t 2 = 0.

s 1 t 2 =- 16 t^2 + 112 t+ 144,

EXAMPLE 5

526 CHAPTER 9 Quadratic Equations, Inequalities, and Functions

NOW TRY
EXERCISE 5
If an object is projected
upward from the top of a
120-ft building at 60 ft per
sec, its position (in feet above
the ground) is given by

,

where tis time in seconds
after it was projected.
When does it hit the ground
(to the nearest tenth)?

s 1 t 2 =- 16 t^2 + 60 t+ 120

NOW TRY
EXERCISE 6
Refer to Example 6.

(a)Use the model to approxi-
mate the CPI for 2005, to
the nearest whole number.

(b)In what year did the CPI
reach 500? (Round down
for the year.)

NOW TRY ANSWERS

5. 5.2 sec after it is projected


6. (a) 578 (b) 1998

NOW TRY

Using a Quadratic Function to Model the CPI

The Consumer Price Index (CPI) is used to measure

trends in prices for a “basket” of goods purchased by

typical American families. This index uses a base year

of 1967, which means that the index number for 1967 is

100. The quadratic function defined by

approximates the CPI for the years 1980–2005, where

xis the number of years that have elapsed since 1980.

(Source:Bureau of Labor Statistics.)

(a)Use the model to approximate the CPI for 1995.

For 1995, so find

Given model

Let

Nearest whole number

The CPI for 1995 was about 456.

(b)In what year did the CPI reach 550?

Find the value of xthat makes

Given model

Let

Standard form

or

Rounding the first solution 22.6 down, the CPI first reached 550 in

2002. (Reject the solution as this corresponds to a year far beyond the

period covered by the model.)

xL205.1,

1980 + 22 =

xL22.6 xL205.1

x=

- 14.8 2 14.8^2 - 41 - 0.065 21 - 3012

21 - 0.065 2

0 =-0.065x^2 + 14.8x- 301

550 =-0.065x^2 + 14.8x+ 249 ƒ 1 x 2 =550.

ƒ 1 x 2 =-0.065x^2 + 14.8x+ 249

ƒ 1 x 2 =550.

ƒ 1152 L 456

ƒ 1152 =-0.065 11522 + 14.8 1152 + 249 x=15.

ƒ 1 x 2 =-0.065x^2 + 14.8x+ 249

x= 1995 - 1980 = 15, ƒ 1152.

ƒ 1 x 2 =-0.065x^2 +14.8x+ 249

EXAMPLE 6

Use

and in the

quadratic formula.

c=- 301

a=-0.065, b=14.8,

NOW TRY