Here are some facts about semi-trucks and a trip between Chicago and New Orleans

Question

Here are some facts about semi-trucks and a trip between Chicago and New Orleans.

(a) The trip is 750 miles.

(b) Running at 50 mph, the truck gets around 4 miles per gallon.

(c) For each mph increase in speed, the truck losses 1/10 of a mile per gallon in mileage.

(d) The driver team gets $27 per hour.

(e) Keeping the truck on the road costs 15 dollars per hour over and above the cost of fuel.

(f) Diesel fuel costs $3.90 per gallon.

Write a function, C(x), for the total cost of driving the truck from Chicago to New Orleans in terms of the constant speed x. At what (constant) speed should the truck be driven to minimize the cost of the trip? Use calculus to minimize costs.

Explanation / Answer

The key to this is buried in the data
The total time spent on the road is:
= ( distance traveled ) / ( constant speed )
= 750/x

The mi/gal that the truck gets is:

4-(x-50)*(1/10)

I can say:
( $ / gal ) / ( mi / gal ) = $ / mi

= 3.9/(4-(x-50)*(1/10))

( $ / mi )*( mi ) = $
= 750*3.9/(4-(x-50)*(1/10))

Now I can say:
C(x) = 750*3.9/(4-(x-50)*(1/10)) + (27+15)*(750/x)

C(x) = 2925/(4-(x-50/10)) + 42*(750/x)

C(x) = 2925/(4-(x-50/10))+(31500/x)

Multiply top and bottom of first term by 10

C(x) = 2925/(40-x+50))+(31500/x)

C(x) = 2925/(90-x)+(31500/x)

This checks out for x=50

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