A truck gets 377/x mpg when driven at a constant speed of x mph (between 25 and 75 mph). If the price of fuel is $1 per gallon and the driver is paid $8 per hour, at what speed between 25 and 75 mph is it most economical to drive? (Give your answer correct to the nearest full mph)
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To determine the most economical speed for the truck to drive, we need to consider both the fuel cost and the driver's cost.
First, let's focus on the fuel cost. The truck gets 377/x miles per gallon when driven at a constant speed of x mph. This means that for every gallon of fuel, the truck can travel 377/x miles.
The price of fuel is $1 per gallon. So, the cost per mile is equal to the fuel cost divided by the distance traveled: 1 / (377/x) = x / 377.
Next, let's consider the driver's cost. The driver is paid $8 per hour. The time it takes to travel a distance at a constant speed is distance divided by speed: 377/x / x = 377 / x^2.
The cost per mile for the driver is equal to the driver's cost divided by the distance traveled: 8 / (377 / x^2) = 8x^2 / 377.
To determine the total cost per mile (combining both fuel and driver costs), we can add the cost per mile for fuel and the cost per mile for the driver:
Total cost per mile = (x / 377) + (8x^2 / 377).
To find the speed at which it is most economical to drive, we need to find the minimum value of the total cost per mile. We can do this by taking the first derivative of the total cost per mile with respect to x, setting it equal to zero, and solving for x.
d/dx [(x / 377) + (8x^2 / 377)] = 0.
Simplifying this equation, we get:
1 / 377 + (16x / 377) = 0.
Combining like terms, we have:
(16x + 1) / 377 = 0.
Solving for x, we get:
16x + 1 = 0.
16x = -1.
x = -1 / 16.
Since x represents the speed, which cannot be negative, we conclude that there is no minimum value for x within the given range of 25 to 75 mph.
Therefore, there is no specific speed within the given range that is most economical to drive the truck.
cost=8*time+ x/(377/x)*$1/gallon*time
x is in miles/hr, t is in time hrs
d cost/dtime=0=8+ 377/x
solve for x, the most economical speed to drive.
x=377/8= ???