To find the balance in an account after 6 years with a continuous compounding interest rate, we use the formula for continuous compounding:
[tex]\[ F = P \cdot e^{(r \cdot t)} \][/tex]
where:
- [tex]\( P \)[/tex] is the principal amount initially deposited.
- [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.
- [tex]\( e \)[/tex] is the base of the natural logarithm (approximately equal to 2.71828).
Let's substitute the given values into the formula:
- [tex]\( P = \$550 \)[/tex]
- [tex]\( r = 8.5\% = \frac{8.5}{100} = 0.085 \)[/tex]
- [tex]\( t = 6 \)[/tex] years
So we have:
[tex]\[ F = 550 \cdot e^{(0.085 \cdot 6)} \][/tex]
Now we compute the exponent part:
[tex]\[ 0.085 \cdot 6 = 0.51 \][/tex]
Next, we apply the exponential function [tex]\( e^{0.51} \)[/tex]:
[tex]\[ F = 550 \cdot e^{0.51} \][/tex]
Calculating the value of [tex]\( e^{0.51} \)[/tex]:
[tex]\[ e^{0.51} \approx 1.665 \][/tex]
Continuing with the calculation:
[tex]\[ F = 550 \cdot 1.665 \approx 915.9101572202375 \][/tex]
Finally, we round this result to the nearest cent:
[tex]\[ F \approx 915.91 \][/tex]
So, the balance in the account after 6 years, rounded to the nearest cent, is:
[tex]\[
F = \$915.91
\][/tex]
