Solve the problem.
-A country has a population growth rate of 2.4% compounded continuously. At this rate, how long will it take for the population of the country to double? Round your answer to the nearest tenth.
A) .29 years
B) 30 years
C) 2.9 years
D) 28.9 years

Question

Quizplus · Accepted Answer

We can use the formula for continuous compounding to solve this problem, which is:

A = Pe^(rt)

Where A is the final amount, P is the initial amount, e is the mathematical constant roughly equal to 2.71828, r is the annual interest rate (or growth rate), and t is the time in years.

We want to know how long it will take for the population to double, which means the final amount A will be twice the initial amount P. So we can write:

2P = Pe^(0.024t)

Simplifying, we can divide both sides by P:

2 = e^(0.024t)

Taking the natural logarithm of both sides, we get:

ln(2) = 0.024t

Solving for t, we get:

t = ln(2)/0.024 ≈ 28.9 years

So it will take about 28.9 years for the population to double, which is the closest choice to our answer. The correct answer is D. We can use the continuous compound interest formula, which has the same form as the population growth formula:
A = Pe^(rt)
where A is the final amount, P is the initial amount, e is the base of the natural logarithm, r is the growth rate, and t is time.
If we let A = 2P (since we want to know when the population doubles), P = 1 (since we are using a relative growth rate), and r = 0.024 (since the growth rate is 2.4%), we can solve for t:
2 = e^(0.024t)
ln(2) = 0.024t
t = ln(2)/0.024 ≈ 28.9 years

Solve the Problem. -A Country Has a Population Growth Rate of 2.4% Compounded