To find the investment amount after 7 years using the continuous growth function \( A(t) = Pe^{rt} \), where:
- \( P \) is the principal amount (initial investment),
- \( r \) is the annual interest rate (as a decimal),
- \( t \) is the time in years,
- \( e \) is the base of the natural logarithm (approximately 2.71828).
In this scenario:
- \( P = 3700 \)
- \( r = 4.5% = 0.045 \)
- \( t = 7 \)
Now, plug these values into the formula:
\[ A(7) = 3700 \cdot e^{0.045 \cdot 7} \]
Calculate \( 0.045 \cdot 7 \):
\[ 0.045 \cdot 7 = 0.315 \]
Next, calculate \( e^{0.315} \):
\[ e^{0.315} \approx 1.3708 \quad (\text{using a calculator}) \]
Now, multiply this by the initial investment:
\[ A(7) = 3700 \cdot 1.3708 \approx 5071.96 \]
So, the investment amount after 7 years will be approximately $5,072.
Among the provided responses, the closest amount is:
$5,069.96