Question

The population of Kingsfield grew from 7,500 to 9,000

in one year. During the same time the population of
Queensville dropped from 32,000 to 25,600. Let the
original populations represent year 1. If these percentage
rates of decline and growth continue, during what year will
Kingsfield have a larger population than Queensville?

Answer 1

year 4

Explanation:

You want to know when a population of 7500 that grows to 9000 in year one will exceed a population of 32,000 that declines to 25,600 in year one if the exponential changes continue at the same rate.

Change rate

The growth factor for the population of Kingsfield is ...

9000/7500 = 1.2 . . . . . a 20% per year growth rate

The growth factor for the population of Queensville is ...

25,600/32,000 = 0.8 . . . . a 20% per year decay rate

Exponential model

Given that the starting populations are defined for year 1, the populations can be modeled as ...

k(t) = 7500·1.2^(t-1)

q(t) = 32000·0.8^(t-1)

We want to know when the populations are equal, so we want to solve ...

k(t) = q(t)

7500·1.2^(t-1) = 32000·0.8^(t-1)

Solution

Dividing by the left side gives ...

1 = (32000/7500)·(0.8/1.2)^(t-1)

Taking logarithms, we have ...

0 = log(320/75) +(t -1)·log(8/12)

t = log(64/15)/log(3/2) +1 ≈ 4.578

The population of Kingsfield will exceed that of Queensville during year 4.

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