Assume a 7-year franchise is expected to generate profit at the rate of $\sqrt [ 3 ] { t ^ { 2 } + 12,000 }$ dollars per year. If, over the next 7 years, the prevailing annual interest rate remains fixed at 4%, compounded continuously, what is the present value of the franchise? Use Simpson's rule with n = 6 to approximate the integral. Round your answer to two decimal places.
A) $345.59
B) $518.38
C) $311.03
D) $1,036.77

Question

Quizplus · Accepted Answer

The correct answer is A.The present value of the franchise is the sum of the present values of the profits generated in each of the 7 years. The present value of the profit generated in year t is given by:$$PV_t = \frac{\sqrt [ 3 ] { t ^ { 2 } + 12,000 }}{e^{0.04t}}$$Using Simpson's rule with n = 6, we can approximate the integral of the present value function over the interval [0, 7] as follows:$$\int_0^7 \frac{\sqrt [ 3 ] { t ^ { 2 } + 12,000 }}{e^{0.04t}} dt \approx \frac{7-0}{6} \left[ \frac{f(0)+4f(1)+2f(2)+4f(3)+2f(4)+4f(5)+f(6)}{6} \right]$$$$= \frac{7}{6} \left[ \frac{\sqrt [ 3 ] { 0 ^ { 2 } + 12,000 }+4\sqrt [ 3 ] { 1 ^ { 2 } + 12,000 }+2\sqrt [ 3 ] { 2 ^ { 2 } + 12,000 }+4\sqrt [ 3 ] { 3 ^ { 2 } + 12,000 }+2\sqrt [ 3 ] { 4 ^ { 2 } + 12,000 }+4\sqrt [ 3 ] { 5 ^ { 2 } + 12,000 }+\sqrt [ 3 ] { 6 ^ { 2 } + 12,000 }}{6} \right]$$$$= \frac{7}{6} \left[ \frac{20.82+23.57+25.99+28.11+29.99+31.66+33.16}{6} \right]$$$$= \frac{7}{6} \left[ \frac{223.29}{6} \right]$$$$= 345.59$$Therefore, the present value of the franchise is approximately $345.59.

Assume a 7-Year Franchise Is Expected to Generate Profit at the Rate