To solve for the future value of an investment compounded continuously, we use the formula for continuous compounding:
[tex]\[ A = P e^{rt} \][/tex]
Where:
- [tex]\( A \)[/tex] is the amount of money accumulated after n years, including interest.
- [tex]\( P \)[/tex] is the principal amount (initial investment).
- [tex]\( e \)[/tex] is the base of the natural logarithm, approximately equal to 2.71828.
- [tex]\( r \)[/tex] is the annual interest rate (expressed as a decimal).
- [tex]\( t \)[/tex] is the time the money is invested for (in years).
Given:
- Principal amount, [tex]\( P = 32000 \)[/tex] dollars
- Annual interest rate, [tex]\( r = 0.07 \)[/tex]
- Time, [tex]\( t = 14 \)[/tex] years
Let's plug these values into the formula and solve step-by-step:
1. Calculate the exponent [tex]\( rt \)[/tex]:
[tex]\[ rt = 0.07 \times 14 = 0.98 \][/tex]
2. Raise [tex]\( e \)[/tex] to the power of 0.98:
[tex]\[ e^{0.98} \approx 2.664456 \][/tex]
3. Multiply the principal amount by this result:
[tex]\[ A = 32000 \times 2.664456 \approx 85222.60 \][/tex]
To find the amount to the nearest dollar, we round [tex]\( 85222.60 \)[/tex] to the nearest whole number:
[tex]\[ A \approx 85223 \][/tex]
Therefore, the amount of money in the account after 14 years, to the nearest dollar, would be $85,223.
