Full solution

Q. Suppose that

\$8000

is invested and pays

4.5\%

per year under the following compounding options. Determine the total amount in the account after

5

years with each option.

\newline

a. Compounded annually

\newline

b. Compounded monthly

\newline

c. Compounded continuously

Calculate Total Amount Annually: Calculate the total amount for the investment compounded annually. $\newline$ We use the formula for compound interest: $A = P(1 + r/n)^{(nt)}$ , where: $\newline$ $A$ = the amount of money accumulated after $n$ years, including interest. $\newline$ $P$ = the principal amount (the initial amount of money). $\newline$ $r$ = the annual interest rate (decimal). $\newline$ $n$ = the number of times that interest is compounded per year. $\newline$ $t$ = the time the money is invested for, in years. $\newline$ For annual compounding, $n = 1$ , so the formula simplifies to $A = P(1 + r)^t$ . $\newline$ $P = \$8000$ , $A$ $0$ or $A$ $1$ , $A$ $2$ years. $\newline$ $A$ $3$ $\newline$ $A$ $4$ $\newline$ $A$ $5$ $\newline$ $A$ $6$
Calculate Total Amount Monthly: Calculate the total amount for the investment compounded monthly. $\newline$ For monthly compounding, $n = 12$ . $\newline$ $P = \$(8000)$ , $r = 4.5\%$ or $0.045$ , $t = 5$ years. $\newline$ $A = 8000(1 + 0.045/12)^{(12*5)}$ $\newline$ $A = 8000(1 + 0.00375)^{60}$ $\newline$ $A = 8000(1.00375)^{60}$ $\newline$ $A = 8000 \times 1.252256$ $\newline$ $A = \$(10018.05)$
Calculate Total Amount Continuously: Calculate the total amount for the investment compounded continuously. $\newline$ We use the formula for continuous compounding: $A = Pe^{rt}$ , where: $\newline$ $e$ = the base of the natural logarithm, approximately equal to $2.71828$ . $\newline$ $r$ = the annual interest rate (decimal). $\newline$ $t$ = the time the money is invested for, in years. $\newline$ $P = \$8000$ , $r = 4.5\%$ or $0.045$ , $t = 5$ years. $\newline$ $A = 8000 \times e^{0.045\times5}$ $\newline$ $e$ $0$ $\newline$ $e$ $1$ $\newline$ $e$ $2$ $\newline$ $e$ $3$