Question

A population grows at the rate of 8% per year. How long does it take for the population to double?

• A. \( 1 \times \log(2) \) years
• B. \( \frac{25}{2} \times \log(2) \) years
• C. 10 years
• D. 12.5 years

Hint:

To calculate the time for a population to double, use the exponential growth formula and solve for \( t \) when \( P(t) = 2P_0 \).

Answer 1

The Correct Option is D

Step 1: Use the formula for exponential growth.
The population growth formula is given by: \[ P(t) = P_0 e^{rt} \] Where \( r \) is the growth rate, \( P_0 \) is the initial population, and \( t \) is the time in years. Step 2: Solve for time.
To find when the population doubles, set \( P(t) = 2P_0 \): \[ 2P_0 = P_0 e^{0.08t} \] \[ 2 = e^{0.08t} \] Taking the natural logarithm of both sides: \[ \ln 2 = 0.08t \quad \Rightarrow \quad t = \frac{\ln 2}{0.08} \] \[ t = \frac{0.693}{0.08} = 12.5 \, \text{years} \] Final Answer: \[ \boxed{12.5 \, \text{years}} \]