Expert: Alon Mandes - 10/21/2009

Question
The cost of operating a truck is 250 + .31v cents per mile when the truck runs at a steady speed of v miles per hour.  The top speed of the truck is 100 mph.  Assume that the driver is paid $14 per hour to drive the truck and he is to begin a 2700 mile trip.
Find the cost of operating the truck in dollars, as a function of the speed v. Find the cost of the driver's wages in dollars, as a function of speed v.  Write the total cost of the trip, as a function of speed v, which is sum of the cost of operating the truck and the cost of the drivers wages.  Then, using analytic methods, determine the speed of the truck that minimizes total cost.

Answer
Let's set C to represent the coast of operating the truck. Let's set S to represent the coast
to be paid to the driver.
1st of all we know that 1$=100Cents , thus : 250 + .31v cents per mile IS (250 + .31v)/100
dollars per mile.

Let's write C as function of v :
-------------------------------
C(v)=(250 + .31v)/100 = (2.5+0.0031) $/mile .
The whole journey is 2700 mile. Therefore :
C(v) = (2.5+0.0031) $/mile * 2700 mile
C(v) = 6750+8.37v $


Let's write S as function of v :
-------------------------------
S=14 $/hour . We need to know how much hours the 2700 miles journey will take :
If we divide 2700/v we get the amount of hours. Thus,
S(v)=14$/hour * 2700/v
S(v)=37800/v $

Now let's write the total coast :
TOT=C(v)+S(v) = 6750+8.37v + 37800/v .
To minimize TOT we will derive , & set the derivative equals zero :
TOT'=8.31-(37800/v²)
TOT'=0 --> 8.31-(37800/v²)=0 --> 8.31=(37800/v²) --> v=21 mile/hour .

Alon.