To determine how much \[tex]$200 invested at 6% interest compounded annually will be worth after 6 years, we can use the compound interest formula given:
\[
A(t) = P \left(1 + \frac{r}{n} \right)^{n t}
\]
where:
- \(P\) is the initial principal balance (\$[/tex]200),
- [tex]\(r\)[/tex] is the annual interest rate (6% or 0.06),
- [tex]\(n\)[/tex] is the number of times interest is compounded per year (annually, so [tex]\(n = 1\)[/tex]),
- [tex]\(t\)[/tex] is the number of years the money is invested (6 years).
Let's substitute these values into the formula:
[tex]\[
A(t) = 200 \left(1 + \frac{0.06}{1} \right)^{1 \cdot 6}
\][/tex]
Simplify the expressions inside the parentheses and the exponent:
[tex]\[
A(t) = 200 \left(1 + 0.06 \right)^6
\][/tex]
[tex]\[
A(t) = 200 \left(1.06 \right)^6
\][/tex]
Now we calculate [tex]\(1.06^6\)[/tex]:
[tex]\[
1.06^6 \approx 1.418519
\][/tex]
Then multiply by the initial principal:
[tex]\[
A(t) \approx 200 \times 1.418519
\][/tex]
[tex]\[
A(t) \approx 283.703822
\][/tex]
Thus, the exact amount after 6 years is approximately \[tex]$283.703822.
When rounding this to the nearest cent, we consider the digit in the third decimal place:
\[
283.70_3\ldots \approx 283.70
\]
So, to the nearest cent, the investment will be worth \$[/tex]283.70 after 6 years.
