To determine the future value of Tyra's investment after 3 years in a savings account with 4.7% annual interest compounded semi-annually, we use the compound interest formula. The formula for compound interest is given by:
[tex]\[ A = P \left(1 + \frac{r}{n} \right)^{nt} \][/tex]
where:
- [tex]\( A \)[/tex] is the amount of money accumulated after n years, including interest.
- [tex]\( P \)[/tex] is the principal amount (the initial sum of money) which is [tex]$5200.
- \( r \) is the annual interest rate (decimal) which is 4.7% or 0.047.
- \( n \) is the number of times interest is compounded per year, which is 2 (since it's compounded semi-annually).
- \( t \) is the time the money is invested for in years, which is 3 years.
Let's plug in the values into the compound interest formula step-by-step:
1. Convert the annual interest rate from a percentage to a decimal:
\[ r = \frac{4.7}{100} = 0.047 \]
2. Identify that the interest is compounded semi-annually, so:
\[ n = 2 \]
3. Insert the principal amount:
\[ P = 5200 \]
4. Time in years is:
\[ t = 3 \]
Now, put the values into the formula:
\[ A = 5200 \left(1 + \frac{0.047}{2} \right)^{2 \cdot 3} \]
5. Simplify inside the parentheses first:
\[ \frac{0.047}{2} = 0.0235 \]
6. Add 1 to the result inside the parentheses:
\[ 1 + 0.0235 = 1.0235 \]
7. Raise this to the power of \( nt \):
\[ (1.0235)^{6} \]
8. Multiply the principal by this result:
\[ A = 5200 \times (1.0235)^6 \]
After performing the calculation, the future value \( A \) comes out to be approximately:
\[ A \approx 5977.65 \]
Thus, the value of Tyra's investment after 3 years, rounded to the nearest cent, will be \$[/tex]5977.65.
