Question

The population of a town increases at a rate proportional to the population at that time. If the population increases from 40 thousands to 60 thousands in 40 years, what will be the population in another 20 years?

(Given: `sqrt(3/2)= 1.2247)`

Answer 1

Step 1: Use the Exponential Growth Model

P(t) = P₀​e^kt

where:

• P = Population at time t
• P₀ = Initial population
• k = Growth constant
• t = Time in years

Step 2: Find k Using Given Data

We are given:

• P₀ = 40,000 
• P(40) = 60,000
• t = 40 years

60,000 = 40,000e^40k

`(60,000)/(40,000) = e^(40k)`

1.5 = e^40k

Taking the natural logarithm:

ln (1.5) = 40k

`k = ln(1.5)/40`

Using ln⁡ (1.5) = 0.4055

`k = 0.4055/40 = 0.01014`

Step 3: Find Population After 60 Years

Now, we need to find P(60)

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

P(60) = 40,000e^60×0.01014

P(60) = 40,000e^0.6084

P(60) = 40,000 × 1.837

P(60) ≈ 73,480