What is the nominal annual rate of interest compounded monthly at which $\$ 1500.00$ will accumulate to $\$ 2836.56$ in eight years and one month? The nominal annual rate of interest is $\square \%$. (Round the final answer to four decimal places as needed. Round all intermediate values to six decimal places as needed.)
Solution
Step 1 ：Given that the accumulated amount (A) is $2836.56, the principal amount (P) is $1500.00, the number of times that interest is compounded per year (n) is 12, and the time the money is invested for in years (t) is 8.0833 years, we need to find the annual interest rate (r).
Step 2 ：We can use the formula for compound interest, rearranged to solve for r: \(r = n[(A/P)^{1/nt} - 1]\).
Step 3 ：Substitute the given values into the formula: \(r = 12[(2836.56/1500.00)^{1/(12*8.0833)} - 1]\).
Step 4 ：Solving the equation gives \(r = 0.0790795796543895\) in decimal form.
Step 5 ：To convert the decimal to a percentage, multiply by 100: \(r_{\text{percent}} = 0.0790795796543895 * 100 = 7.90795796543895\).
Step 6 ：Round the percentage to four decimal places to get the final answer: \(r_{\text{rounded}} = 7.908\).
Step 7 ：Final Answer: The nominal annual rate of interest compounded monthly at which $1500.00 will accumulate to $2836.56 in eight years and one month is \(\boxed{7.908\%}\).
