Engineering Econ Please show step by step so i can use as reference for my answe
ID: 1155697 • Letter: E
Question
Engineering Econ Please show step by step so i can use as reference for my answers.
(15) 2. 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 10 to 40 horsepower. The motor will have an annual ownership cost of $120 plus $0.60 per horsepower. The operating costs of such motors for 1 horsepower-hour will be $0.055 divided by the horsepower rating. Nine thousand horsepower-hours will be needed per year. (a) Determine what horsepower should be specified for a minimum total annual cost. (b) Show that you have minimized total cost per year.Explanation / Answer
Total Cost Function=120+0.6x+9000(0.055)/x
Differentiatinng with respect to x to find FOC and equaling it to 0 we get
dTC/dx=0.6-495/x^2=0
0.6x^2=495
x^2=495/0.6
x=28.72 horsepowers
Now lets find whether this value provides maximum or minimum cost hence differentiaiting equation 1 again we get
d^2TC/dx^2=0.6-495/x^2=0.99/x^3>0 therefore x=28.72 is minimum value of horsepower that provides minimum cost
x=28.72 horsepowers needed for minimum cost
Total Cost Function=120+0.6x+9000(0.055)/x=120+0.6(28.72)+(495/28.72)=$154.47
x1=30 then cost=120+0.6(30)+(495/30)=$154.5
x2=25 then cost=120+0.6(25)+(495/25)=$154.8
x2<x<x1 and TC2>TC<TC1
Hence proved
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