Answer :
To determine the future value of an investment that is compounded continuously, we can use the formula for continuous compounding:
[tex]\[ A = P e^{rt} \][/tex]
where:
- [tex]\( A \)[/tex] is the amount of money accumulated after [tex]\( t \)[/tex] years, including interest.
- [tex]\( P \)[/tex] is the principal amount (the initial amount of money, which is [tex]$25,000 in this case). - \( r \) is the annual interest rate (expressed as a decimal, so 2.65% becomes 0.0265). - \( t \) is the time in years (since it is not specified in your question, we will assume it to be 5 years). - \( e \) is the base of the natural logarithm, approximately equal to 2.71828. Given these parameters: - \( P = 25,000 \) - \( r = 0.0265 \) - \( t = 5 \) Substituting these into the formula: \[ A = 25,000 \times e^{0.0265 \times 5} \] First, we calculate the exponent: \[ 0.0265 \times 5 = 0.1325 \] Next, we evaluate \( e^{0.1325} \): \[ e^{0.1325} \approx 1.141898 \] Now we multiply this by the principal amount: \[ A = 25,000 \times 1.141898 \] \[ A \approx 28,541.98 \] Therefore, the future value of the investment after 5 years, rounded to the nearest penny, is $[/tex]28,541.98.
[tex]\[ A = P e^{rt} \][/tex]
where:
- [tex]\( A \)[/tex] is the amount of money accumulated after [tex]\( t \)[/tex] years, including interest.
- [tex]\( P \)[/tex] is the principal amount (the initial amount of money, which is [tex]$25,000 in this case). - \( r \) is the annual interest rate (expressed as a decimal, so 2.65% becomes 0.0265). - \( t \) is the time in years (since it is not specified in your question, we will assume it to be 5 years). - \( e \) is the base of the natural logarithm, approximately equal to 2.71828. Given these parameters: - \( P = 25,000 \) - \( r = 0.0265 \) - \( t = 5 \) Substituting these into the formula: \[ A = 25,000 \times e^{0.0265 \times 5} \] First, we calculate the exponent: \[ 0.0265 \times 5 = 0.1325 \] Next, we evaluate \( e^{0.1325} \): \[ e^{0.1325} \approx 1.141898 \] Now we multiply this by the principal amount: \[ A = 25,000 \times 1.141898 \] \[ A \approx 28,541.98 \] Therefore, the future value of the investment after 5 years, rounded to the nearest penny, is $[/tex]28,541.98.