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A truck gets 500/x miles per gallon (mpg) when driven at a constant speed of x mph, where 40 ≤ x ≤ 80. If the price of fuel is $2.80/gal and the driver is paid $10/hr, at what speed is it most economical for the trucker to drive? (Round your answer to two decimal places.)

User Snappy
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1 Answer

0 votes

Answer :

42.26 mph (approx)

Explanation :

Given,

Fuel consumption = 500/x miles per gallon,

Speed = x miles per hour

Also, hourly cost = $2.80/gal,

So, fuel cost =
(1)/((500)/(x))* 2.80


=(2.80x)/(500) dollar per mile,

The driver earns $10 per hour.

So, labour cost = (10 dollars per hour) × (1/x hr per mile)


=(10)/(x) dollars per mile,

Thus, total cost,


P = (10)/(x) + (2.8x)/(500)

Differentiating with respect to x


(dP)/(dx) = -(10)/(x^2) + (2.8)/(500)

Again differentiating with respect to x,


(d^2P)/(dx^2)=(30)/(x^3)

When,


(dP)/(dt) = 0


\implies 2.8 x^2 = 5000


\implies x^2 = 1785.71428


\implies x = 42.26

At x = 42.26,
(d^2P)/(dx^2) = positive.

Hence, the speed would be 42.26 miles per hour.

User Duncan Malashock
by
8.0k points
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