Question

The population of a town grows at a rate proportional to its size. If it grows from 40,000 to 60,000 in 40 years, what will it be in another 20 years?

Hint:

For exponential growth problems, find the growth factor first, then raise it to the required time ratio.

Answer 1

Concept: Growth proportional to population follows exponential growth: \[ \frac{dP}{dt} = kP \] Solution: \[ P(t) = P_0 e^{kt} \] We use given values to find \(k\), then predict future population.
Step 1: Given data \[ P_0 = 40000, \quad P(40) = 60000 \] Using: \[ P(40) = 40000 e^{40k} \] \[ 60000 = 40000 e^{40k} \] \[ \frac{3}{2} = e^{40k} \] \[ k = \frac{1}{40} \ln\left(\frac{3}{2}\right) \]
Step 2: Population after another 20 years Total time = 60 years \[ P(60) = 40000 e^{60k} \] Substitute \(k\): \[ P(60) = 40000 \left(e^{40k}\right)^{3/2} \] Since \(e^{40k} = \frac{3}{2}\), \[ P(60) = 40000 \left(\frac{3}{2}\right)^{3/2} \]
Step 3: Simplify \[ \left(\frac{3}{2}\right)^{3/2} = \sqrt{\frac{27}{8}} \] \[ P(60) = 40000 \cdot \sqrt{\frac{27}{8}} \] Approximate: \[ \sqrt{\frac{27}{8}} \approx 1.837 \] \[ P(60) \approx 40000 \times 1.837 \approx 73,480 \] Final Answer: \[ \boxed{P \approx 73,500 \text{ (approximately)}} \] Explanation: Exponential growth means the growth factor remains constant. Using the growth ratio from the first 40 years allows prediction of future population.