To determine how much should be invested now to have [tex]$1500 in three years at an interest rate of 5% per year, compounded continuously, we can use the formula for continuous compounding present value. Here’s a detailed, step-by-step solution:
Step 1: Identify the variables.
- Future value (FV) or final amount: \( \$[/tex]1500 \)
- Annual interest rate (r): [tex]\( 5\% = 0.05 \)[/tex]
- Time (t): [tex]\( 3 \)[/tex] years
Step 2: Use the continuous compounding present value formula.
The present value (PV) is given by the formula:
[tex]\[ PV = \frac{FV}{e^{rt}} \][/tex]
where [tex]\( e \)[/tex] is the base of the natural logarithm, approximately equal to 2.71828.
Step 3: Substitute the known values into the formula.
[tex]\[ PV = \frac{1500}{e^{(0.05 \times 3)}} \][/tex]
Step 4: Calculate the exponent.
[tex]\[ 0.05 \times 3 = 0.15 \][/tex]
Step 5: Calculate [tex]\( e^{0.15} \)[/tex].
[tex]\[ e^{0.15} \approx 1.16183424 \][/tex] (without rounding intermediate computations)
Step 6: Divide the future value by this result to get the present value.
[tex]\[ PV = \frac{1500}{1.16183424} \][/tex]
Step 7: Perform the division.
[tex]\[ PV \approx 1291.060 \][/tex]
Step 8: Round the present value to the nearest cent.
[tex]\[ PV \approx 1291.06 \][/tex]
Therefore, to have [tex]$1500 in three years at an interest rate of 5% per year, compounded continuously, you should invest approximately $[/tex]1291.06 now.
