Exponential Growth Math Example 2

Follow the full solution, then compare it with the other examples linked below.

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A city's population grows at

3\%

per year. If the current population is

200{,}000

, when will it reach

300{,}000

1
Model: $P(t) = 200000(1.03)^t$ . Set $P(t) = 300000$ : $(1.03)^t = 1.5$ .
2
Take $\ln$ : $t \cdot \ln(1.03) = \ln(1.5)$ .
3
$t = \frac{\ln(1.5)}{\ln(1.03)} = \frac{0.4055}{0.02956} \approx 13.72$ years.

Answer

t \approx 13.7 \text{ years}

Exponential growth with a constant percentage rate uses the model

P(t) = P_0(1+r)^t

. Solving for time requires logarithms. The Rule of 70 estimates doubling time as

70/r

(here

70/3 \approx 23

years), but this problem asks when the population reaches

1.5

times its initial value, which takes less time.

About Exponential Growth

Exponential growth occurs when a quantity increases by a constant multiplicative factor over equal intervals.

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