Question

The population of a city is modeled by the equation P(t)=426,474e0.25t where t is measured in years. If the city continues to grow at this rate, how many years will it take for the population to reach one million? Round your answer to the nearest year. The population will reach one million in years.

Answer 1

3 years

Explanation:

You want t such that P(t) = 426474·e^(0.25t) will be one million.

Logarithms

Logarithms are used to transform an exponential equation into a linear equation. To make the solution easier to evaluate, we can do some preliminary work.

P(t) = 1,000,000

426,474·e^(0.25t) = 1,000,000 . . . . . . use the desired value of P(t)

e^(0.25t) = 1,000,000/426,474 . . . . . . divide by 426474

0.25t = ln(1,000,000/426,474) . . . . . . take natural logs

t = 4·ln(1/0.426747) = -4·ln(0.426474) ≈ 3.4 ≈ 3

The population will reach one million is about 3 years.

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