Questions: Question 8 (1 point) The population of a small village is now 807, which is 3 times what it was 5 years ago. If the population continues to increase exponentially at this rate, then the population 7 years from now will be 3757 1560 2905 4842

Solution Steps

To solve this problem, we need to determine the exponential growth rate of the population and then use it to predict the population 7 years from now.

1. Let the population 5 years ago be \( P_0 \).
2. Given that the current population \( P \) is 807, and it is 3 times what it was 5 years ago, we can write \( 807 = 3 \times P_0 \).
3. Solve for \( P_0 \).
4. Use the exponential growth formula \( P(t) = P_0 \times e^{kt} \) to find the growth rate \( k \).
5. Use the growth rate \( k \) to predict the population 7 years from now.

Given that the current population \( P = 807 \) is 3 times the population 5 years ago, we can express this as: \[ P_0 = \frac{P}{3} = \frac{807}{3} = 269 \]

Using the exponential growth formula \( P = P_0 e^{kt} \), we can find the growth rate \( k \). From the equation: \[ 807 = 269 e^{5k} \] we can rearrange it to: \[ e^{5k} = \frac{807}{269} = 3 \] Taking the natural logarithm of both sides gives: \[ 5k = \ln(3) \implies k = \frac{\ln(3)}{5} \approx 0.2197 \]

To find the population 7 years from now, we use the formula again: \[ P(t) = P_0 e^{kt} \] where \( t = 5 + 7 = 12 \): \[ P(12) = 269 e^{12k} = 269 e^{12 \cdot 0.2197} \approx 3757.0181 \]

Final Answer