Find the required annual interest rate, to the nearest tenth of a percent, for $\$ 13,236$ to grow to $\$ 18,651$ if interest is compounded semiannually for 6 years.
A. $11.6 \%$
B. $5.8 \%$
C. $2.9 \%$
D. $8.7 \%$
Answer
\(\boxed{5.8\%}\) is the required annual interest rate, to the nearest tenth of a percent, for $13,236 to grow to $18,651 if interest is compounded semiannually for 6 years.
Step 1 ：We are given that the principal amount (P) is $13,236, the final amount (A) is $18,651, the interest is compounded semiannually so n = 2, and the time (t) is 6 years. We need to find the annual interest rate (r).
Step 2 ：We can use the formula for compound interest, which is A = P(1 + r/n)^(nt), and rearrange it to solve for r: r = n[(A/P)^(1/nt) - 1].
Step 3 ：Substituting the given values into the formula, we get r = 2[(18651/13236)^(1/(2*6)) - 1].
Step 4 ：Solving this equation gives r = 0.05798454647993889.
Step 5 ：Converting this decimal to a percentage and rounding to the nearest tenth of a percent gives r = 5.8%.
Step 6 ：\(\boxed{5.8\%}\) is the required annual interest rate, to the nearest tenth of a percent, for $13,236 to grow to $18,651 if interest is compounded semiannually for 6 years.
