To determine how much Bill will have in his account after four years with continuously compounded interest, we use the formula for continuously compounded interest, which is given by:
[tex]\[ A = P \cdot e^{(rt)} \][/tex]
where:
- [tex]\( A \)[/tex] is the amount of money accumulated after the time period.
- [tex]\( P \)[/tex] is the principal amount (initial deposit).
- [tex]\( r \)[/tex] is the annual interest rate (as a decimal).
- [tex]\( t \)[/tex] is the time the money is invested for (in years).
- [tex]\( e \)[/tex] is the base of the natural logarithm (approximately equal to 2.71828).
In this specific case:
- [tex]\( P = 3300 \)[/tex] (dollars).
- [tex]\( r = 0.065 \)[/tex] (6.5% annual interest rate).
- [tex]\( t = 4 \)[/tex] (years).
Let's plug these values into the formula:
[tex]\[ A = 3300 \cdot e^{(0.065 \cdot 4)} \][/tex]
First, we calculate the exponent:
[tex]\[ 0.065 \cdot 4 = 0.26 \][/tex]
Now, we calculate [tex]\( e^{0.26} \)[/tex]:
[tex]\[ e^{0.26} \approx 1.2969300866657718 \][/tex] (using a calculator)
Next, we multiply this result by the principal amount:
[tex]\[ A = 3300 \cdot 1.2969300866657718 \][/tex]
[tex]\[ A \approx 4279.869285997047 \][/tex]
Finally, we round this amount to the nearest cent:
[tex]\[ A \approx 4279.87 \][/tex]
So, after four years, Bill will have approximately $4279.87 in his account, assuming he makes no withdrawals.
