Answer:
As you know it would take approximately 27.96 years for the investment of $5000 to grow to $8200 at an interest rate of 7.5% per year, compounded quarterly
Explanation:
To find the time required for the investment to grow to $8200, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = final amount ($8200 in this case)
P = principal amount ($5000 in this case)
r = annual interest rate (7.5% or 0.075 in decimal form)
n = number of times interest is compounded per year (quarterly in this case)
t = time in years (unknown)
Rearranging the formula to solve for t:
t = (log(A/P) / (n * log(1 + r/n)))
Plugging in the values, we get:
t = (log(8200/5000) / (4 * log(1 + 0.075/4)))
Using a calculator, let's find the value of t:
t ≈ (log(1.64) / (4 * log(1.01875)))
t ≈ (0.5132 / (4 * 0.00459))
t ≈ (0.5132 / 0.01836)
t ≈ 27.96
Therefore, it would take approximately 27.96 years for the investment of $5000 to grow to $8200 at an interest rate of 7.5% per year, compounded quarterly.