Question A population of deer experienced \( 8.2 \% \) monthly growth compounded continuously for \( m \) months. At the population had doubled in size. The doubling-time for this population of deer is \( m \) months. Solve for \( m \) rounding to the nearest whole month. Provide your answer below:

**Step 1:** The continuous compound growth formula is \[ P(m) = P_0 \, e^{rm}, \] where \( P_0 \) is the initial population, \( r \) is the growth rate, and \( m \) is the time in months. **Step 2:** Since the population doubled, we have \[ P(m) = 2P_0. \] Substitute into the formula: \[ 2P_0 = P_0 \, e^{rm}. \] **Step 3:** Dividing both sides by \( P_0 \) (assuming \( P_0 \neq 0 \)) gives: \[ 2 = e^{rm}. \] **Step 4:** Taking the natural logarithm on both sides yields: \[ \ln(2) = rm. \] Solve for \( m \): \[ m = \frac{\ln(2)}{r}. \] **Step 5:** Given the monthly growth rate \( r = 8.2\% = 0.082 \), substitute into the expression for \( m \): \[ m = \frac{\ln(2)}{0.082}. \] **Step 6:** Using the approximation \( \ln(2) \approx 0.693 \), calculate \( m \): \[ m \approx \frac{0.693}{0.082} \approx 8.445. \] **Step 7:** Rounding \( 8.445 \) to the nearest whole month gives \( m \approx 8 \). **Final Answer:** \( 8 \) months.