To determine the balance in a money market account after 32 years with an initial deposit of [tex]$5000 and an annual interest rate of 7.6% compounded monthly, follow these steps:
1. Initial Deposit (Principal, \( P \)):
- The initial amount of money deposited is \( P = \$[/tex]5000 \).
2. Annual Interest Rate ([tex]\( r \)[/tex]):
- The annual interest rate is given as 7.6%. To use it in calculations, convert this percentage to a decimal: [tex]\( r = \frac{7.6}{100} = 0.076 \)[/tex].
3. Number of Times Interest is Compounded per Year ([tex]\( n \)[/tex]):
- The interest is compounded monthly, so [tex]\( n = 12 \)[/tex] (months in a year).
4. Number of Years ([tex]\( t \)[/tex]):
- The period over which the money is invested is [tex]\( t = 32 \)[/tex] years.
5. Compound Interest Formula:
- The balance [tex]\( A \)[/tex] in the account can be calculated using the compound interest formula:
[tex]\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\][/tex]
- Substituting the given values into the formula:
[tex]\[
A = 5000 \left(1 + \frac{0.076}{12}\right)^{12 \times 32}
\][/tex]
6. Calculate the Balance:
- First, calculate the monthly interest rate:
[tex]\[
\frac{r}{n} = \frac{0.076}{12} \approx 0.006333
\][/tex]
- Add 1 to the monthly interest rate:
[tex]\[
1 + \frac{r}{n} \approx 1 + 0.006333 = 1.006333
\][/tex]
- Compute the exponent [tex]\( nt \)[/tex]:
[tex]\[
12 \times 32 = 384
\][/tex]
- Raise the base (1.006333) to the power of 384:
[tex]\[
1.006333^{384} \approx 11.294671092258398
\][/tex]
- Multiply by the initial deposit:
[tex]\[
A = 5000 \times 11.294671092258398 \approx 56473.35546129199
\][/tex]
7. Round to the Nearest Cent:
- Finally, round the computed balance to the nearest cent:
[tex]\[
A \approx 56473.36
\][/tex]
Final Balance:
After 32 years, with an initial deposit of [tex]$5000 and an annual interest rate of 7.6% compounded monthly, the balance in the account will be approximately $[/tex]56,473.36 when rounded to the nearest cent.
