Question

The population of a city is given by P(t) = Poe

-0.04t where t is time measured in years and
P0 is the population at time t=0. Assume that Po = 1,000,000.
a. Find the population when t=3.
b. During what year will the population drop below 750,000. (solve an equation, no
guess-and-check)

Answer 1

a. 885,920

b. year 8

Explanation:

The population at a given time is described by the exponential formula ...

P(t) = P0·e^(-0.04t)

where P0 is given as 1,000,000.

a.

We are asked for the value of P(3). This ccan be found by substituting 3 for t in the equation and evaluating the numerical expression.

P(3) = 1,000,000·e^(-0.04·3) = 1,000,000·e^(-0.12)

P(3) ≈ 886,920

The population when t=3 is about 886,920.

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b.

We can put the given numbers in the equation and solve for t.

750,000 = 1,000,000·e^(-0.04t)

0.75 = e^(-0.04t) . . . . . divide by 1,000,000

ln(0.75) = -0.04t . . . . . take the natural log

-ln(0.75)/0.04 = t ≈ 7.192

The population will drop below 750,000 in year 8.

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Additional comment

These values can be confirmed by a graphing calculator.

Note that "year 1" is the year between t=0 and t=1. So, "year 8" is the year between t=7 and t=8.