To solve for the future value of an investment that is being compounded continuously, we can use the formula for continuously compounded interest:
\[ A = Pe^{rt} \]
Here:
- \( A \) is the amount of money accumulated after n years, including interest.
- \( P \) is the principal amount (the initial amount of money).
- \( r \) is the annual interest rate (in decimal form).
- \( t \) is the time the money is invested for in years.
- \( e \) is the base of the natural logarithm, approximately equal to 2.71828.
Given the following:
- \( P = \$380 \) (initial amount invested)
- \( r = 6.2\% = 0.062 \) (interest rate converted to decimal form)
- \( t = 15 \) years (time)
We plug these values into our formula:
\[ A = 380e^{0.062 \times 15} \]
Now we need to calculate the value using the base of the natural logarithm. This calculation can be done on a scientific calculator or by using mathematical software that can handle the natural exponential function.
\[ A = 380 \times e^{0.93} \]
\[ A = 380 \times 2.534... \] (value of \( e^{0.93} \) approximated with a calculator)
\[ A \approx 963.16 \]
The last step is to round the amount to the nearest ten dollars.
Rounding $963.16 to the nearest ten dollars gives us $960.
So, after 15 years, with an interest rate of 6.2% compounded continuously, Jace's investment would be approximately $960 to the nearest ten dollars.
