Problema Solution
A truck driving 260 miles on a flat interstate highway at a constant rate of 50 miles per hour gets 7 miles to the gallon. For each mile per hour increase in speed, the truck loses a tenth of a mile per gallon in its mileage. Drivers are paid $27.50 per hour in wages, and the costs for running the truck is $11.33 per hour and diesel is $2.49. What constant speed (between 50mph and speed limit of 65mph) should the truck drive to minimize the total cost of the trip?
Answer provided by our tutors
I assume that we need to find the minimum of:
Total Cost = Wage of the driver + Truck fixed cost + Gas cost
Let
t = represent the number of hours driven
x = represent miles per hour increase in speed
The wage of the driver = 27.50t
Truck fixed cost = 11.33t
The cost of the fuel is the gallons used times $2.49.
The amount of gallons used is the total distance traveled divided by your fuel efficiency.
(For example if you traveled 260 miles and the fuel efficiency was 7 miles per gallon, then you would have used 260/7 gallons.)
Your Fuel efficiency is [7 - 0.1x] because for every x increased the fuel efficiency decreases by 0.1.
So total amount of gallons used = 260 / (7 - 0.1x)
So,
Gas cost = 2.49*(260 / (7 - 0.1x))
So far the total cost equation is:
Total Cost = = 27.50t + 11.33t + 2.49*(260 / (7 - 0.1x))
Now we know that speed is distance divided by time follows time is distance divided by speed:
t = 260/(50 + x)
Plug the equation for t in the Total Cost formula:
Total Cost = 27.50(260/(50 + x)) + 11.33(260/(50 + x)) + 2.49*(260 / (7 - 0.1x))
If we
graph the equation
y = 27.50(260/(50 + x)) + 11.33(260/(50 + x)) + 2.49*(260 / (7 - 0.1x))
we see that as x increases (x = 0 to x = 15) the total cost decreases (since the speed is 50 mph when x = 0 and the speed is 65 mph when x = 15).
So to minimize the total cost the speed of the truck should be 65 mph.