To find out how much Mandy needs to deposit as principal in a savings account that earns continuous interest to reach a goal of $800 in one year, we can use the formula for continuous compounding:
\[ A = Pe^{rt} \]
Where:
- \( A \) is the amount of money accumulated after n years, including interest ($800 in this case),
- \( P \) is the principal amount (the initial amount of money),
- \( r \) is the annual interest rate (decimal),
- \( t \) is the time the money is invested for in years,
- \( e \) is the base of the natural logarithm (approximately 2.71828).
We need to solve for \( P \):
\[ 800 = Pe^{0.15 \times 1} \]
Calculating \( e^{0.15} \):
\[ e^{0.15} \approx 1.161834 \]
Now, plug this value back into the equation:
\[ 800 = P \times 1.161834 \]
Now, solve for \( P \):
\[ P = \frac{800}{1.161834} \]
Calculating the value of \( P \):
\[ P \approx \frac{800}{1.161834} \approx 688.29 \]
Thus, rounding to the nearest cent, Mandy needs to deposit approximately $688.29.
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