To find the investment amount after 9 years using the continuous change function, we can use the formula:
\[ A(t) = P \cdot e^{rt} \]
Where:
- \( A(t) \) is the amount of money accumulated after time \( t \),
- \( P \) is the principal amount (the 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 equal to 2.71828).
Given:
- \( P = 10,500 \)
- \( r = 0.0375 \) (since 3.75% = 0.0375 in decimal)
- \( t = 9 \)
Now, plug the values into the formula:
\[ A(9) = 10,500 \cdot e^{0.0375 \cdot 9} \]
First, calculate \( 0.0375 \cdot 9 = 0.3375 \).
Next, calculate \( e^{0.3375} \):
Using a calculator or software, \( e^{0.3375} \approx 1.40243 \).
Now multiply it by the principal:
\[ A(9) \approx 10,500 \cdot 1.40243 \approx 14,715.00 \]
So, the investment amount after 9 years will be approximately \( \text{A}(9) \approx 14,715.00 \).
Thus, the correct answer is:
$14,715.12