Write a function for the total cost of driving a truck from Chicago to New Orlea

Question

Write a function for the total cost of driving a truck from Chicago to New Orleanse with respect to the average speed. Use calculus to minimize cost. Please explain the reasoning.

The parpenters are:

1. The trip is 750 miles

2. running at 50mph the truck gets 4 mpg

3. for each mpg increase in speed the truck losses 1/10th of a mile per gallon in mileage.

4. for each mph decrease the truck gains 1/10 of a mile per gallon in mileage.

5 the driver gets paid $27 per hour

6. Kepping the truck on the road cost $15 hour over and above the cost of fuel.

7. Fuel cost $3.90 per gallon.

Explanation / Answer

Let Speed (S) = (50 + X) mph

// Given Mileage at 50mph is 4mpg and will looses 1/10 mpg with 1mph increase in speed

.: Mileage (M) = (4 - X/10) mpg

.: Travel (T) = 750/(50 + X) hr

// Given Drive cost = $27/hr, Vehicle cost = $15/hr and Fuel cost = $3.90/g

.: Cost C(X) = {27*T} + {15*T} + {3.90*750/M}

C(X) = 750[ 42/(50+X) + 39/(40-X) ] -----------------------> eq.1

If C'(X) = 0 then at that X C(X) can be maximum or minimum.

From the eq.1, we can understand that when X tends to 40 or -50 , Cost tends to Infinite. So there can be only be minimum value of C(X) at X other than 40 or -50.

.: C'(X) = 0

720[ -42/(50 + X)2 + 39/(40 - X)2] = 0

39(50 + X)2 = 42(40 - X)2

3X2 - 7260X - 30300 = 0

.: X = 2424.1664 or - 4.1664

As truck cannot go at speed of (50 + 2424.1664) it is (50 - 4.1664)

C(-4.1664) = $1295.5554

.: Minimum Cost = $1295.5554 at truck speed (50 - 4.1664) = 45.8336 mph

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