Questions: A population doubles in size every 39 years. Assuming exponential growth, find the (a) Annual growth rate (b) Continuous growth rate Give your answers as a percentage, rounded to three decimal places. The annual growth rate is i %. The continuous growth rate is i %.

Solution Steps

To solve this problem, we need to use the properties of exponential growth.

(a) For the annual growth rate, we can use the formula for exponential growth: \( P(t) = P_0 \times (1 + r)^t \). Given that the population doubles in 39 years, we can set up the equation \( 2 = (1 + r)^{39} \) and solve for \( r \).

(b) For the continuous growth rate, we use the formula \( P(t) = P_0 \times e^{kt} \). Given that the population doubles in 39 years, we can set up the equation \( 2 = e^{39k} \) and solve for \( k \).

Given that the population doubles every 39 years, we use the formula for exponential growth: \[ 2 = (1 + r)^{39} \] Solving for \( r \): \[ r = 2^{\frac{1}{39}} - 1 \] The annual growth rate \( r \) is approximately \( 0.01793 \) or \( 1.793\% \).

For continuous growth, we use the formula: \[ 2 = e^{39k} \] Solving for \( k \): \[ k = \frac{\ln(2)}{39} \] The continuous growth rate \( k \) is approximately \( 0.01777 \) or \( 1.777\% \).

Final Answer