Final answer:
To find the number of years it will take for $7500 to accumulate to $9517.39 at 3% compounded semiannually, we can use the compound interest formula and solve for t. Substituting the given values into the formula, we find that t ≈ 8.44 years.
Step-by-step explanation:
To find the number of years it will take for $7500 to accumulate to $9517.39 at 3% compounded semiannually, we can use the compound interest formula:
A = P(1 + r/n)^(nt)
Where:
- A is the final amount
- P is the principal amount
- r is the annual interest rate (as a decimal)
- n is the number of times interest is compounded per year
- t is the number of years
Given that P = $7500, A = $9517.39, r = 3% (converted to decimal form as 0.03), and n = 2 (compounded semiannually), we can rearrange the formula to solve for t:
t = log(A/P) / (n * log(1 + r/n))
Substituting the given values into the formula, we get:
t = log(9517.39/7500) / (2 * log(1 + 0.03/2))
Using a calculator, we find that t ≈ 8.44 years.