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The population of a town grows at a rate proportional to the population present at time t. The initial population of 500 increases by 25% in 10 years. What will be the population in 40 years

User Mpg
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1 Answer

2 votes

Answer:

The population in 40 years will be 1220.

Explanation:

The population of a town grows at a rate proportional to the population present at time t.

This means that:


P(t) = P(0)e^(rt)

In which P(t) is the population after t years, P(0) is the initial population and r is the growth rate.

The initial population of 500 increases by 25% in 10 years.

This means that
P(0) = 500, P(10) = 1.25*500 = 625

We apply this to the equation and find t.


P(t) = P(0)e^(rt)


625 = 500e^(10r)


e^(10r) = (625)/(500)


e^(10r) = 1.25

Applying ln to both sides


\ln{e^(10r)} = ln(1.25)


10r = ln(1.25)


r = (ln(1.25))/(10)


r = 0.0223

So


P(t) = 500e^(0.0223t)

What will be the population in 40 years

This is P(40).


P(t) = 500e^(0.0223t)


P(40) = 500e^(0.0223*40) = 1220

The population in 40 years will be 1220.

User Yehor Nemov
by
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