To find out how much \[tex]$730 will grow to at an interest rate of 5% per year, compounded daily for 7 years, we will use the formula for compound interest. Here are the steps you need to follow:
1. Identify the variables in the compound interest formula:
- Principal (\(P\)): \$[/tex]730
- Annual interest rate ([tex]\(r\)[/tex]): 5%, or 0.05 in decimal form
- Number of times the interest is compounded per year ([tex]\(n\)[/tex]): 365 (daily)
- Number of years ([tex]\(t\)[/tex]): 7
2. Write down the compound interest formula:
[tex]\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\][/tex]
Here, [tex]\(A\)[/tex] is the amount of money accumulated after n years, including interest.
3. Substitute the variables into the compound interest formula:
[tex]\[
A = 730 \left(1 + \frac{0.05}{365}\right)^{365 \times 7}
\][/tex]
4. Calculate the exponent:
[tex]\[
365 \times 7 = 2555
\][/tex]
5. Calculate the rate divided by the number of times interest is compounded:
[tex]\[
\frac{0.05}{365} = 0.0001369863 \approx 0.000137
\][/tex]
6. Add 1 to the fraction:
[tex]\[
1 + 0.000137 = 1.000137
\][/tex]
7. Raise 1.000137 to the power of 2555:
[tex]\[
1.000137^{2555} \approx 1.41958
\][/tex]
8. Multiply the principal by this result:
[tex]\[
A = 730 \times 1.41958 \approx 1035.8944793561636
\][/tex]
9. Round the final amount to the nearest cent:
[tex]\[
A \approx 1035.89
\][/tex]
So, \[tex]$730 will grow to \$[/tex]1035.89 after 7 years at an annual interest rate of 5% compounded daily.
