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

Given:
The population grows at a rate proportional to its size.
Initial population P₀ = 40,000
Population after 40 years = 60,000

Step 1: Form the Differential Equation
Since growth is proportional to population:

dP/dt = kP

where k is the constant of proportionality.

Step 2: Solve the Differential Equation

dP/P = k dt

Integrating:

ln P = kt + C

P = Ce^kt

At t = 0, P = 40,000

So,

40,000 = Ce⁰ ⇒ C = 40,000

Therefore,

P = 40,000 e^kt

Step 3: Use Given Data to Find k

At t = 40 years, P = 60,000

60,000 = 40,000 e^40k

60,000 / 40,000 = e^40k

3/2 = e^40k

Taking logarithm:

40k = ln(3/2)

k = (1/40) ln(3/2)

Step 4: Find Population After Another 20 Years

Total time = 60 years

P(60) = 40,000 e^60k

= 40,000 e^{(60/40) ln(3/2)}

= 40,000 e^{(3/2) ln(3/2)}

= 40,000 (3/2)^3/2

(3/2)^3/2 = √((3/2)³) = √(27/8)

= √3.375 ≈ 1.837

Therefore,

P ≈ 40,000 × 1.837

≈ 73,480

Final Answer:
The population after another 20 years will be approximately 73,480.