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rate of cost of fuel FC = price of fuel P * volume of fuel consumed V

rate of cost of fuel FC is proportional to the square of speed. So volume V of fuel consumed per hour is proportional to the square of speed v in kmph.

rate of fuel cost FC = $ 64 /hour for Speed v = 16 kmph FC = K v² , where K is the constant of proportionality. => $ 64 = K * 16² => K = 0.25 $-Hr/km²

rate of other costs OC = $ 100 / hour

total costs rate = TC = FC + OC = 0.25 v² + 100 $ / hour

Let the engine in the vehicle drive the vehicle for a distance of S km. Then the time taken to cover the distance S is : T = S / v. The total distance S to be travelled is a constant.

Total cost for T hours in $ C = T * (0.25 v² + 100) = S / v * (0.25 v² + 100) = 0.25 S v + 100 S / v

differentiate C wrt v and find the value of v for which C is minimum.

d C / d v = 0.25 S - 100 S / v² = 0 for minimum so 0.25 = 100 / v² v² = 400 v = 20 kmph =========== So the minimum cost of running the engine is : 0.25 * 20² + 100 = 200 $ / hour