A design engineer must select the horsepower for a motor to drive a pump. Horsep

Question

A design engineer must select the horsepower for a motor to drive a pump. Horsepower rating is a continuous design parameter with values in the range of 30 to 60 horsepower. The motor will have an annual ownership cost of $130 plus $0.30 per horsepower. The operating costs of such motors for one horsepower-hour will be $0.085 divided by the horsepower rating. 8,000 horsepower hours will be needed per year. a. Determine what horsepower should be specified for a minimum total annual cost. b. How do you know that you have minimized total cost per year? a. To minimize total annual cost, the horsepower for the motor should be horsepower. (Round to the nearest whole number.) Find the optimal value of horsepower at which minimum total annual cost occurs by taking the first derivative of the equation for total annual cost with respect to horsepower rating and setting it equal to zero. Recall that total annual cost is the sum of the fixed cost, directly varying cost, and indirectly varying cost. Form: Total Cost-ax b/xk

Explanation / Answer

a) Total Cost Function=130 + 0.3x + 8000(0.085)/x

Differentiatinng with respect to x to find FOC and equating it to 0 we get

d(TC)/dx= 0.3 - 680/x2

0.3 - 680/x2 = 0

680 =0.3x2

x = 47.61 = 48 horespower

To minimize the total annual cost, the horsepower for the motor should be 48 horsepower

b) Now lets find that this value provides minimum cost or not

to find this we find the SOC ( second order condition)

hence differentiating d(TC)/dx again we get

d2(TC)/dx2 = 1360 / x3

Put X = 48, we get 0.012 > 0

therefore x= 48 is value of horsepower that provides minimum cost.

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