Answer:
7.64%
Explanation:
To find the annual interest rate, compounded annually, for an investment to grow from $500 to $2005.76 in 13 years, we can use the formula for compound interest:
\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]
Where:
- A = the amount of money accumulated after n years, including interest.
- P = the principal amount ($500 in this case).
- r = the annual interest rate (to be calculated).
- n = the number of times that interest is compounded per year (1 for annually).
- t = the time the money is invested for, in years (13 years in this case).
We want to find the annual interest rate (r). Rearranging the formula to solve for r, we get:
\[ r = n \left(\left(\frac{A}{P}\right)^{\frac{1}{nt}} - 1\right) \]
Substituting the given values:
- A = $2005.76
- P = $500
- n = 1 (compounded annually)
- t = 13 years
\[ r = 1 \left(\left(\frac{2005.76}{500}\right)^{\frac{1}{1*13}} - 1\right) \]
\[ r = \left(\left(4.01152\right)^{\frac{1}{13}} - 1\right) \]
\[ r ≈ \left(1.0764 - 1\right) \]
\[ r ≈ 0.0764 \]
So, the annual interest rate, compounded annually, for the investment to grow to $2005.76 in 13 years is approximately 7.64%.
