A company uses a truck to deliver the products. To estimate costs, the manager m

Question

A company uses a truck to deliver the products. To estimate costs, the manager models the gas consumption by the function G(x)=1/250(1,200/x+x) gal/mile, assuming the truck is driven by a constant speed of x miles per hour, for x ge 5. The driver is paid $20 per hour to drive the truck 250 miles, and gasoline costs $1.90 per gallon. Find an expression for the total cost C(x) of the trip, and use C(x) to determine if the costs is increasing or decreasing when the truck is driven 40 miles per hour, 50 miles per hour and 60 miles per hour. Interpret the results.

Explanation / Answer

G(x) = (1/250)(1200/x + x) [gal/mi]
x is speed in mph, x 5

C(x) = 20*t + 1.90*G(x)*x*t, where t is the time

Let's say the total distance traveled is d = 250 = x*t. d is constant. Let's substitute t = d/x.

C(x) = 20*d/x + 1.90*G(x)*d

C(x) = 20*250/x + 1.90*(1/250)*(1200/x + x)*250

C(x) = 5000/x + 1.90*(1200/x + x)

To see if the cost is increasing or decreasing, we must take the derivative. Plugging in the values for x will give us the slope at those points and tell us if the cost, C(x) is increasing or decreasing for those speeds.

C'(x) = -5000/x2 + 1.90(-1200/x2 + 1)

C'(40) = -5000/1600 + 1.90(-1200/1600 + 1) = -2.65

C'(50) = -1.01

C'(60) = -0.122

The cost decreases, but at declining rates, as the speed increases. The optimal speed with respect to reducing cost is greater than 60 mph. The optimal speed is 61.9 mph.

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