Final answer:
To find the nominal interest rate compounded quarterly, we use the compound interest formula A = P(1 + r/n)^(nt). By plugging in the values and solving for r, we determine that the rate that turns P12,000 into P15,000 over 8 years, with quarterly compounding, is 3%.
Step-by-step explanation:
The student is asking to determine the nominal interest rate compounded quarterly that would cause an initial deposit of P12,000 to grow to P15,000 over a span of 8 years. To solve this, we use the compound interest formula, which is A = P(1 + r/n)^(nt). In this formula, A is the amount of money accumulated after n years, including interest, P is the principal amount (the initial amount of money), r is the annual interest rate (decimal), n is the number of times that interest is compounded per year, and t is the time the money is invested or borrowed for, in years.
In this case, P = 12,000, A = 15,000, n = 4 (since the interest is compounded quarterly), and t = 8. We want to find the value of r. Rearranging the formula to solve for r, we have (A/P) = (1 + r/n)^(nt), and therefore, (15,000/12,000) = (1 + r/4)^(4*8).
Calculating for r, we find (1 + r/4)^32 = 1.25. Solving for r involves extracting the 32nd root of 1.25 and then multiplying by 4 after subtracting 1, to convert from the quarterly rate to the nominal annual rate. After performing the calculation, we find the nominal rate that satisfies this equation is 3%, which corresponds to option b.