Find the amount that results from the given investment.
$\$ 400$ invested at $6 \%$ compounded quarterly after a period of $4 \frac{1}{2}$ years
After $4 \frac{1}{2}$ years, the investment results in $\$ \square$.
(Round to the nearest cent as needed.)
Answer
Rounding to the nearest cent, the future value of the investment after 4.5 years is \(\boxed{522.94}\)
Step 1 ：Given that the principal amount (P) is $400, the annual interest rate (r) is 6% or 0.06 in decimal, the number of times that interest is compounded per year (n) is 4 (quarterly compounding), and the time the money is invested for (t) is 4.5 years.
Step 2 ：We can use the formula for compound interest to find the future value of the investment. The formula is: \(A = P (1 + \frac{r}{n})^{nt}\), where A is the amount of money accumulated after n years, including interest.
Step 3 ：Substituting the given values into the formula, we get: \(A = 400 (1 + \frac{0.06}{4})^{4*4.5}\)
Step 4 ：Solving the equation, we find that \(A = 522.9362543328123\)
Step 5 ：Rounding to the nearest cent, the future value of the investment after 4.5 years is \(\boxed{522.94}\)
