To find out how much Manny's investment will be worth after 20 years with continuous compounding, we can use the formula:
\[ A = Pe^{rt} \]
Where:
- \( A \) is the amount accrued (future value).
- \( P \) is the principal amount (initial investment).
- \( r \) is the annual interest rate (in decimal).
- \( t \) is the time in years.
- \( e \) is the base of the natural logarithm, approximately equal to 2.71828.
In this case:
- \( P = 100 \) (the initial investment)
- \( r = 0.02 \) (2% as a decimal)
- \( t = 20 \) (the number of years)
Now we can plug in the values:
\[ A = 100 e^{0.02 \times 20} \] \[ A = 100 e^{0.4} \]
Next, we need to calculate \( e^{0.4} \). Using a calculator:
\[ e^{0.4} \approx 1.49182 \]
Now we can calculate \( A \):
\[ A = 100 \times 1.49182 \approx 149.182 \]
Finally, rounding to the nearest dollar:
\[ A \approx 149 \]
Thus, after 20 years, Manny’s investment will be worth approximately $149.