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The fixed cost for a steam line per foot of pipe is CF = $475X +$75 per year. The cost for loss of heat from the pipe is CHEAT = $4.75/(X"^) per year. Here, X represents the thickness of insulation in feet, and X is a continuous design variable. a.What is the optimal thickness of insulation? b. Show why your answer in part (a) minimizes your total cost per year. c. What kind of trade-off are you making considering the cost of insulation for the steamline? d. Calculate the total cost of a 7-mile pipeline with 1-foot thick insulation.

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