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Question

The population of a city is modeled by the equation P(t)=262,576e^0.25t 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 hundredth of a year (i.e. 2 decimal places).

 

The population will reach one million in______   years.

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Answer 1

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P(t)=262,576e^0.25t

1,000,000 = 262,576e^0.25t

Taking natural logarithms of both sides gives:

ln(1,000,000) = ln(262,576e^0.25t)

ln(1,000,000) = ln(262,576) + ln(e^0.25t)

ln(1,000,000) = ln(262,576) + 0.25t

t = [ln(1,000,000) - ln(262,576)] / 0.25

t =5.35 years

Answer 2

1000000/262576 = e^(0.25 t) =>

ln(1000000/262576) = 0.25 t =>

ln(3.8084) = 0.25 t =>

t = 5.35 years.