You are the owner of a delivery company with one truck. The trucks fuel economy is 1200/x miles per gallon when driving at x miles per hour (where 80 >= x <= 110). The price of fuel is $1 per gallon. You pay the truck driver $8 per hour.
If a delivery 100 miles away is planned, at what speed between 80 and 110 mph should the truck to drive to minimize the cost of the trip?
I'm having trouble finding the equation to differentiate.
Please show work.
figure the cost of the trip: driver pay + fuel cost
if it's 100 miles, and the driver drives at x miles/hr, then the trip takes 100/x hours.
fuel used is x/1200 gal/mi * 100 mi
cost = 8 * 100/x + 1 * 100 * x/1200
Looks like dcost/dx = 0 at about x=97mph
Good luck finding a highway with that speed limit!
To solve this problem, we need to determine the speed at which the truck should drive to minimize the cost of the trip. We can start by setting up an equation for the total cost of the trip.
Let's define:
v = speed at which the truck drives (in miles per hour)
d = distance of the trip (in miles)
f = fuel economy at the speed v (in miles per gallon)
p = price of fuel per gallon (in dollars)
h = time taken for the trip (in hours)
r = rate of pay for the truck driver (in dollars per hour)
The total cost of the trip can be calculated as follows:
Cost = (d / f) * p + h * r
Given that the distance of the trip is 100 miles, the price of fuel is $1 per gallon, and the rate of pay for the truck driver is $8 per hour, we can simplify the equation as:
Cost = (100 / f) * 1 + h * 8
Now, let's express the fuel economy as a function of the speed v. It is mentioned that the fuel economy is 1200/x miles per gallon when driving at x miles per hour. We can denote this as:
f = 1200 / v
Substituting this into the cost equation, we get:
Cost = (100 / (1200/v)) + h * 8
Next, we need to express the time taken for the trip as a function of the speed v. The time taken can be calculated as:
h = d / v
Substituting this into the cost equation, we get:
Cost = (100 / (1200/v)) + (100 / v) * 8
Now, our task is to find the speed v that minimizes the cost. To do this, we differentiate the cost equation with respect to v and set it equal to zero:
d(Cost) / dv = -100 * (1200 / v^2) + 800 / v^2 = 0
Simplifying the equation:
-120000 / v^2 + 800 / v^2 = 0
Combining like terms:
-120000 + 800 = 0
-119200 = 0
This equation does not have a solution, indicating that there is no minimum cost within the given speed range (80 <= v <= 110). Therefore, we need to reconsider the problem or the given parameters of the truck's fuel economy and driver's rate of pay.
It's important to note that this conclusion should be checked with the company's constraints and requirements, as well as considering factors like road safety regulations and potential penalties for exceeding speed limits.