Question: Suppose that $83,000 is invested at \(4\frac{1}{2}\%\) interest, compounded quarterly. a) Find…

Suppose that $83,000 is invested at \(4\frac{1}{2}\%\) interest, compounded quarterly.

a) Find the function for the amount to which the investment grows after \(t\) years.

b) Graph the function.

c) Find the amount of money in the account at \(t = 0\), 4, 7, and 10 years.

d) When will the amount of money in the account reach $100,000?

a) The function for the amount to which the investment grows after \(t\) years is \(A(t) = \boxed{\quad}\).

(Simplify your answer. Type an expression using \(t\) as the variable.)

Solution

Suppose that $83,000 is invested at \(4 \frac{1}{2}\%\) interest, compounded quarterly. a) Find the function for the amount to which the investment grows after \(t\) years. The formula for compound interest is: \[ A(t) = P \left(1 + \frac{r}{n}\right)^{nt} \] where: - \(P = 83000\) is the principal, - \(r = 0.045\) is the annual interest rate (4.5%), - \(n = 4\) is the number of times the interest is compounded per year, - \(t\) is the number of years. Substitute the known values into the formula: \[ A(t) = 83000 \left(1 + \frac{0.045}{4}\right)^{4t} \] Simplifying inside the parentheses: \[ A(t) = 83000 \left(1 + 0.01125\right)^{4t} \] \[ A(t) = 83000 (1.01125)^{4t} \]