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# Find the most economical speed for a truck driver subject to constraints

A truck driver must drive 300 miles at a constant velocity of miles per hour, where speed laws require . The truck consumes fuel at the rate of gallons per hour, and the cost of the fuel is 0.3 dollars per gallon. The driver is paid dollars per hour. Find the speed at which the total cost of the trip is minimized when

1. ,
2. ,
3. ,
4. ,
5. .

Before dealing with specific values of , we write down an equation for the total cost of the trip in terms of the speed . The driver must be paid dollars per hour, and the trip takes hours (300 miles divided by the number of miles per hour), so we have Next, the total cost of the fuel is the number of hours of the trip (given by ) times the number of gallons of fuel per hour times the cost of the fuel (30 cents per gallon): So the total cost of the trip in terms of the speed is Now for the specific values of of the problem.

1. If we have Setting this derivative equal to 0 to find the critical points of we have Since Therefore, has a minimum at mph. The cost at this speed 2. If we have Setting this derivative equal to 0 to find the critical points of , we have Since the cost has a minimum at mph. The cost at this speed is then 3. If we have Setting this derivative equal to 0, However, in this case , and the problem tells us that speed laws require . Since the cost is decreasing on the interval (since the derivative is negative everywhere on the interval) we take mph. Then, 4. If , then just as above we take mph and obtain 5. If , we again have mph and get the minimal cost as 