- The investment is calculated using the continuous compounding formula \(A(t) = Pe^{rt}\).
- Given \(P = 10500\), \(r = 0.0375\), and \(t = 9\), we calculate the exponent as \(0.0375 \times 9 = 0.3375\).
- The final amount is \(A(9) = 10500 \times e^{0.3375}\), which yields approximately \(\$14,715.12\).
- Therefore, the investment amount after 9 years is \(\boxed{\$14,715.12}\).
Explanation
- Analysis of the Investment Model
The investment model is governed by the continuous compounding formula, which is given by \(A(t) = Pe^{rt}\). Here, the variables are defined as follows:
- \(P\) is the initial principal investment, which is \(\$10,500\).
- \(r\) is the annual growth rate, which is \(3.75\%\) or \(0.0375\) as a decimal.
- \(t\) is the time period in years, which is \(9\).
Our objective is to calculate the final amount \(A(9)\) by substituting these values into the formula.
- Calculating the Final Amount
We substitute the given values into the equation:
\[A(9) = 10500 \times e^{(0.0375 \times 9)}\]
First, we calculate the exponent:
\[0.0375 \times 9 = 0.3375\]
Next, we evaluate the exponential term:
\[e^{0.3375} \approx 1.4014396\]
Finally, we multiply this by the initial principal of \(\$10,500\):
\[A(9) = 10500 \times 1.4014396 \approx 14715.1159\]
Rounding this result to the nearest cent gives us approximately \(\$14,715.12\).
- Conclusion
Comparing our calculated result of \(\$14,715.12\) with the provided options:
- \(\$14,715.12\)
- \(\$14,781.48\)
- \(\$15,049.96\)
- \(\$16,785.21\)
Our result matches the first option. Thus, the value of the investment after 9 years is \(\boxed{\$14,715.12}\).
Examples
Continuous compounding is a fundamental concept in finance used to determine how money grows when interest is added to an account balance at every possible instant. For example, if you place a sum of money into a high-yield savings account or a retirement fund that compounds continuously, your balance does not just grow annually or monthly, but grows steadily over every moment. This model helps investors predict their future wealth and make informed decisions about long-term savings plans.