A truck has a minimum speed of 11 mph in high gear. When traveling x mph, the tr

Question

A truck has a minimum speed of 11 mph in high gear. When traveling x mph, the truck burns diesel fuel at the rate of 0.0048246 (625/x + x) gal/mile Assuming that the truck cannot be driven over 55 mph and that diesel fuel costs $1.44 a gallon, find the following. The steady speed that will minimize the cost of the fuel for a 370 mile trip. The steady speed that will minimize the total cost of the trip if the driver is paid $12 an hour. The steady speed that will minimize the total cost of a 660 mile trip if the driver is paid $20 an hour.

Explanation / Answer

Solution:

diesel fuel consumption = 0.0048246 ( (625/x) + x ) gal/mile

(a) Cost of fuel C = [0.0048246 ( (625/x) + x )] 1.44 * 370
C = 2.57 ((625/x) + x)
to minimize cost find x for dC/dx = 0
C' = (-625/x^2) + 1 = 0
x^2 = 625
x = 25 mph

(b) C = 2.57 ((625/x) + x) + 12 *370/x
C' = -2.57*(625/x^2) + 2.57 - 12*370/x^2 = 0
x^2 = 2352.63
x = 48.50 mph

(c) C = 0.0048246 ((625/x) + x)* 1.44*660 + 20*660/x
C = 4.585((625/x) + x) + 20*660/x
C' = (-4.585*625/x^2) + 4.585 + (-13200/x^2) = 0
x^2 = 16065.625 / 4.585 = 3503.95
x = 59.19 mph
Since it cannot be driven over 55 mph
x = 55 mph

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