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
-
,
-
,
-
,
-
,
-
.
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.
- 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
- 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
- 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,
- If
, then just as above we take
mph and obtain
- If
, we again have
mph and get the minimal cost as