To determine the amount of money in the investment account after 14 years, given the continuous compounding interest formula [tex]\( V = P e^{rt} \)[/tex], where:
- [tex]\( P \)[/tex] is the initial investment,
- [tex]\( r \)[/tex] is the annual interest rate,
- [tex]\( t \)[/tex] is the number of years,
- [tex]\( e \)[/tex] is the base of the natural logarithm.
Let's use the values provided:
1. The initial investment [tex]\( P = \$6650 \)[/tex].
2. The annual interest rate [tex]\( r = 7.2\% = 0.072 \)[/tex] (converted to a decimal).
3. The number of years [tex]\( t = 14 \)[/tex].
First, we substitute these values into the formula:
[tex]\[ V = 6650 \times e^{0.072 \times 14} \][/tex]
Next, we calculate the exponent:
[tex]\[ 0.072 \times 14 = 1.008 \][/tex]
So, the expression becomes:
[tex]\[ V = 6650 \times e^{1.008} \][/tex]
Using the value of [tex]\( e \)[/tex] (approximately 2.71828), we compute [tex]\( e^{1.008} \)[/tex]:
[tex]\[ e^{1.008} \approx 2.739 \][/tex]
Now multiply the initial investment by this exponential factor:
[tex]\[ V = 6650 \times 2.739 \][/tex]
Perform the multiplication:
[tex]\[ V \approx 18221.77 \][/tex]
Therefore, the amount of money in the account after 14 years, rounded to the nearest cent, is:
[tex]\[ \$18221.77 \][/tex]
