To solve the problem of finding how much a principal of [tex]$5000 will be worth after 14 years when it is invested at an annual interest rate of 5.5%, compounded annually, follow these steps:
1. Understand the formula for compound interest:
\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]
where:
- \( A \) is the amount of money accumulated after \( n \) years, including interest.
- \( P \) is the principal amount (the initial amount of money).
- \( r \) is the annual interest rate (decimal).
- \( n \) is the number of times the interest is compounded per year.
- \( t \) is the number of years the money is invested for.
Since the interest is compounded annually, \( n = 1 \).
2. Plug in the given values:
- \( P = 5000 \)
- \( r = 5.5\% = 0.055 \) (conversion from percentage to decimal)
- \( n = 1 \)
- \( t = 14 \)
3. Substitute the values into the formula:
\[ A = 5000 \left(1 + \frac{0.055}{1}\right)^{1 \times 14} \]
4.Simplify the formula:
\[ A = 5000 \left(1 + 0.055\right)^{14} \]
\[ A = 5000 \left(1.055\right)^{14} \]
5. Calculate the value of \((1.055)^{14}\):
\[ (1.055)^{14} \approx 2.11609146 \]
6. Multiply the result by the principal amount ($[/tex]5000):
[tex]\[ A = 5000 \times 2.11609146 \][/tex]
[tex]\[ A \approx 10580.457309184592 \][/tex]
7. Round the result to the nearest dollar:
[tex]\[ A \approx 10580 \][/tex]
So, the investment will be worth approximately $10,580 after 14 years.
