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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 Nt=N0ekt

(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 Nt=N0ekt, where N0is the initial population, Ntis the current population, t is time and kis positive constant.

03

Part (b) of Step 1. Population is 2009

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

N0=900,000,Nt=800,000,t=2

Substitute the values of N0,Nt,tin the function role="math" localid="1647288977383" Nt=N0ekt

Nt=N0ekt800,000=900,000ek脳289=ek脳2ln89=lnek脳2ln89=k脳2k=ln892k=-0.0639

Now population in 2009 can be calculated by

substituting N0=900,000,k=-0.0639,t=4in the function Nt=N0ekt

Nt=N0ektN2009=900,000e-0.0639脳4N2009=697,007

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