I need help with this please

Question

The population of a city is modeled by the equation P(t) = 278,395e^0.25t

 where t is measured in years. If the city continues to grow at this rate, in approximately how many years will it take for the population to reach one million? (Round your answer to two decimal places.)

Answer 1

Heya Afreen.

So the equation given is meant to tell you the population, P(t), if you input the time in years (t). For example, if you plug in t = 0:

P(t) = 278,395e^0.25t

P(0) = 278,395e^0.25(0)

P(0) = 278,395e⁰

P(0) = 278,395 × 1

P(0) = 278,395

Which means the current population (t = 0 years means now) is 278,395.

Now in the question, they give you instead the population, and ask you to find time. So we can do the same thing, where we plug in the value we know to solve for the variable we don't know.

P(t) = 278,395e^0.25t

1,000,000 = 278,395e^0.25t

From here, we want to isolate t.

1,000,000 = 278,395e^0.25t

1,000,000 / 278,395 = e^0.25t

To reverse exponents, we need to use logarithms. In this specific case, we use ln, which is the logarithm base e:

1,000,000 / 278,395 = e^0.25t

ln(1,000,000 / 278,395) = ln(e^0.25t)

ln(1,000,000 / 278,395) = 0.25t

ln(1,000,000 / 278,395) / 0.25 = t

And there we go. You just need to plug those values into a calculator (although you could have been doing so as we were going through the steps, for example by replacing 1,000,000 / 278,395 with 3.592).

Hope that was helpful. Let me know if you have any other questions.