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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)`
Solution
Step 1: Use the Exponential Growth Model
P(t) = P0ekt
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 = 40,000e40k
`(60,000)/(40,000) = e^(40k)`
1.5 = e40k
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) = 40,000e60k
P(60) = 40,000e60×0.01014
P(60) = 40,000e0.6084
P(60) = 40,000 × 1.837
P(60) ≈ 73,480
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