The cost for operating a commercial truck is knv^(1/2) where k is a constant of

Question

The cost for operating a commercial truck is knv^(1/2) where k is a constant of proportionality, v is velocity in miles per hour, and n is the trip length in miles. It is estimated that at 85 mph, the average cost of operation is $5.2 per mile. The truck owner wants to minimize the cost of operation, to do this he needs to balance the operating cost against the cost of delays and unscheduled maintenance, which is assumed to be $10 per hour. What is the optimal velocity in miles per hour needed to minimize the total costs? (provide answer with one decimal place
precision

Explanation / Answer

5.2=k*1*85^0.5

k=0.564

Operational truck cost/mile=0.564*v^0.5

delay cost/mile=10/v

Total cost/mile=T=0.564*v^0.5+10/v

For minimum total cost,

dT/dv=0

0.282v^-0.5-10v^-2=0

Solving for v,

v=10.80miles/hour

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