Sure! Let's solve this problem step-by-step using the given compound interest formula:
[tex]\[F = P\left(1+\frac{r}{n}\right)^{n t}\][/tex]
Here’s a breakdown of the values given in the problem:
- [tex]\( P \)[/tex] is the principal amount, which is \[tex]$800.
- \( r \) is the annual interest rate, which is 6.5%. Expressed as a decimal, this is \( 0.065 \).
- \( n \) is the number of times interest is compounded per year. Since it is compounded quarterly, \( n = 4 \).
- \( t \) is the time the money is invested for, in years, which is 5 years.
Plugging these values into our compound interest formula:
\[ F = 800 \left(1 + \frac{0.065}{4}\right)^{4 \times 5} \]
First, we calculate \(\frac{r}{n}\):
\[ \frac{0.065}{4} = 0.01625 \]
Next, we add this value to 1:
\[ 1 + 0.01625 = 1.01625 \]
Now we need to raise this value to the power of \( nt \):
\[ 1.01625^{20} \]
Using a calculator, we find that:
\[ 1.01625^{20} \approx 1.380425 \]
Finally, multiply this result by the principal amount \( P \):
\[ F = 800 \times 1.380425 \approx 1104.34 \]
So, the balance after 5 years, rounded to the nearest cent, would be:
\[ \boxed{1104.34} \]
Thus, the balance after 5 years with a 6.5% interest rate, compounded quarterly, would be \$[/tex]1104.34.
