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 most economical speed to drive, we need to determine the cost per mile for each speed between 25 and 75 mph.
1. First, let's calculate the gallons of fuel consumed for each speed. The formula for fuel consumption is:
Fuel consumption (gallons) = distance (miles) / mileage (miles per gallon)
2. Since the truck gets 377/x mpg at a constant speed of x mph, we can substitute the given values to calculate the fuel consumption at each speed. The formula becomes:
Fuel consumption (gallons) = distance (miles) / (377/x)
3. To simplify the expression, we can multiply numerator and denominator by x:
Fuel consumption (gallons) = (distance * x) / 377
4. The cost per mile is the product of the fuel consumption and the price of fuel. The formula is:
Cost per mile = Fuel consumption (gallons) * Price of fuel
5. Given that the price of fuel is $1 per gallon, the cost per mile formula becomes:
Cost per mile = (distance * x) / 377
6. Finally, to find the most economical speed, we need to compare the cost per mile at each speed from 25 to 75 mph. We can calculate the cost per mile for each speed, and the speed with the lowest cost per mile will be the most economical speed to drive.
Let's calculate the cost per mile for each speed and find the most economical one.
To determine the most economical speed to drive the truck, we need to find the speed at which the cost per mile is minimized. The cost per mile is determined by the cost of fuel and the driver's wages.
Let's first calculate the cost of fuel per mile:
Cost of fuel per mile = (1 gallon / 377 miles) × x miles/gallon = x / 377 dollars/mile
Next, let's calculate the driver's wages per mile:
Driver's wages per mile = ($8 per hour) / (x miles/hour) = 8 / x dollars/mile
The total cost per mile is the sum of the cost of fuel per mile and the driver's wages per mile:
Total cost per mile = Cost of fuel per mile + Driver's wages per mile
Total cost per mile = (x / 377) + (8 / x)
To find the most economical speed, we can find the speed at which the total cost per mile is minimized. We can do this by finding the minimum of the function:
f(x) = (x / 377) + (8 / x)
To find the minimum, we can take the derivative of f(x), set it to zero, and solve for x:
f'(x) = (d/dx)(x / 377) + (d/dx)(8 / x)
f'(x) = (1 / 377) - (8 / x^2)
Set f'(x) = 0 and solve for x:
(1 / 377) - (8 / x^2) = 0
(1 / 377) = (8 / x^2)
x^2 = 8 * 377
x^2 = 3016
x = sqrt(3016)
x ≈ 54.96
Since the speed needs to be rounded to the nearest full mph, the most economical speed to drive the truck is approximately 55 mph.
I assume "most economical" means lowest cost per mile.
The driver's wages:
$/mi = $/hr * hr/mi = $/hr / (mi/hr)
The truck:
$/mi = $/gal * gal/mi = ($/gal) / (mi/gal)
If speed is x, and y is cost/hr, including the driver's wages
y = 8/x + 1 * 1/(377/x)
y = 8/x + x/377
For minimum cost/mile,
y' = -8/x^2 + 1/377
x^2 = 3016
x = 54.9