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Precalculus Enhanced with Graphing Utilities

Sullivan

$Math Studyset 91影视 Explanations$ Math

6th Edition 路 1200 pages

ISBN: 9780321795465

Chapter 5: Q 8 (page 335)

Population Decline
The population of a midwestern city follows the exponential law.
(a) If N is the population of the city and t is the time in years, express N as a function of t.
(b) If the population decreased from 900,000 to 800,000 from 2005 to 2007, what will the population be in 2009?

Short Answer

Expert verified

(a) N as a function of t can be expressed as $N (t) = N_{0} e^{k t}$

(b) Population in 2009 will be 697007

Step by step solution

01

Step 1. Given information

The population of a midwestern city follows the exponential law.

02

Part (a) of Step 1. N as a function of t

If N is the population of the city and t is the time in years, then N as a function of t can be expressed as $N (t) = N_{0} e^{k t}$ , where $N_{0}$ is the initial population, $N (t)$ is the current population, t is time and $k$ is positive constant.

03

Part (b) of Step 1. Population is 2009

The population decreased from 900,000 to 800,000 from 2005 to 2007

$N_{0} = 900, 000, N (t) = 800, 000, t = 2$

Substitute the values of $N_{0}, N (t), t$ in the function role="math" localid="1647288977383" $N (t) = N_{0} e^{k t}$

$N (t) = N_{0} e^{k t} 800, 000 = 900, 000 e^{k 脳 2} \frac{8}{9} = e^{k 脳 2} \ln (\frac{8}{9}) = \ln (e^{k 脳 2}) \ln (\frac{8}{9}) = k 脳 2 k = \frac{\ln (\frac{8}{9})}{2} k = - 0.0639$

Now population in 2009 can be calculated by

substituting $N_{0} = 900, 000, k = - 0.0639, t = 4$ in the function $N (t) = N_{0} e^{k t}$

$N (t) = N_{0} e^{k t} N (2009) = 900, 000 e^{- 0.0639 脳 4} N (2009) = 697, 007$

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Most popular questions from this chapter

In Problems $33 - 40$ , the graph of an exponential function is given. Match each graph to one of the following functions.

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