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: Apply the exponential growth model.
The population at time \(t\) is modeled by:\[P(t) = P_0 e^{rt}\]Where:\( r \) = growth rate,\( P_0 \) = initial population,\( t \) = time in years.Step 2: Determine the doubling time.
To find the time it takes for the population to double, set \( P(t) = 2P_0 \):\[2P_0 = P_0 e^{0.08t}\]\[2 = e^{0.08t}\]Applying the natural logarithm:\[\ln 2 = 0.08t \quad \Rightarrow \quad t = \frac{\ln 2}{0.08}\]\[t = \frac{0.693}{0.08} = 12.5 \, \text{years}\] Final Answer: \[ \boxed{12.5 \, \text{years}} \]