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)
To find the speed at which it is most economical to drive the truck, we need to determine the cost per mile for each speed and then choose the speed with the lowest cost per mile.
Let's break down the cost per mile at each speed:
The cost per mile consists of two components: the cost of fuel and the cost of the driver's time.
1. Cost of fuel:
The cost of fuel per mile is calculated by dividing the price per gallon by the truck's miles per gallon (mpg). In this case, the price of fuel is $1 per gallon, so the cost of fuel per mile is 1/mpg.
2. Cost of driver's time:
To calculate the cost of the driver's time per mile, we need to know the time it takes to drive one mile at a given speed. We can find this by dividing the distance traveled (1 mile) by the speed (x mph). The time taken to drive one mile at speed x is 1/x hours. Since the driver is paid $8 per hour, the cost of the driver's time per mile is 8 * (1/x) = 8/x dollars per mile.
Now, let's calculate the total cost per mile for each speed between 25 and 75 mph:
Total cost per mile = Cost of fuel per mile + Cost of driver's time per mile
For each speed x between 25 and 75 mph, we substitute the values into the cost per mile formula and then find the lowest cost per mile.
Let's calculate the cost per mile for a few speeds:
At x = 25 mph:
Cost of fuel per mile = 1 / (377/25) = 25/377 dollars per mile
Cost of driver's time per mile = 8 / 25 = 8/25 dollars per mile
Total cost per mile = (25/377) + (8/25) = (25*8 + 377*8) / (377*25) = 202 / 9425 dollars per mile
Repeat the above calculations for each speed from 26 mph to 75 mph.
At x = 26 mph:
Cost of fuel per mile = 1 / (377/26) = 26/377 dollars per mile
Cost of driver's time per mile = 8 / 26 = 4/13 dollars per mile
Total cost per mile = (26/377) + (4/13) = (26*13 + 377*4) / (377*13) = 502 / 4901 dollars per mile
Repeat the above calculations for each speed from 27 mph to 75 mph.
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Continue this process for each speed from 25 mph to 75 mph, and then calculate the total cost per mile for each speed.
Finally, choose the speed with the lowest total cost per mile as the most economical speed to drive the truck. Round the answer to the nearest full mph.