Question

The formula A = 252e^.049t models the population of a particular city, in thousands, t years after 1998. When will the population of the city reach 373 thousand?

Answer 1

Answer: Approximately the year 2006

Roughly 8 years after 1998

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Work Shown:

A = 373 represents a population of 373 thousand.

Plug in this value of A and solve for t. We'll need natural logs (LN) to isolate the variable.

`A = 252e^(0.049t)\\\\373 = 252e^(0.049t)\\\\373/252 = e^(0.049t)\\\\1.4801587 \approx e^(0.049t)\\\\`

Apply natural logs to both sides.

`\text{Ln}(1.4801587) \approx \text{Ln}\left(e^(0.049t)\right)\\\\\text{Ln}(1.4801587) \approx 0.049t*\text{Ln}\left(e\right)\\\\\text{Ln}(1.4801587) \approx 0.049t*1\\\\\text{Ln}(1.4801587) \approx 0.049t\\\\t \approx \text{Ln}(1.4801587)/0.049\\\\t \approx 8.0030472\\\\`

It takes about 8 years for the population to reach 373 thousand.

Since t = 0 starts at 1998, we get to the year 1998+8 = 2006.