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?
Can someone explain step by step how to do this? I have no idea how to approach it.
Thanks in advance!
cost= 8t+fuel cost.
fuel cost= cost per gallon*gallons needed
= 1*100/(1200/x)
But t= 100miles/x
cost= 8*100/x + x/12 check that
dcost/dx=0=-800/x^2 + 1/12
x^2=9600
x= you do it.
how do you check?
To solve this problem, we need to find the speed between 80 and 110 mph at which the cost of the trip is minimum.
Step 1: Calculate the fuel consumption at each speed from 80 to 110 mph.
The fuel economy is given by 1200/x miles per gallon, where x is the speed in mph. So, to find the fuel consumption at each speed, we divide the distance (100 miles) by the fuel economy (1200/x) to get the number of gallons required for the trip at each speed.
Step 2: Calculate the cost of fuel at each speed.
The price of fuel is $1 per gallon. Multiply the number of gallons required for the trip by the price per gallon to find the cost of fuel at each speed.
Step 3: Calculate the time taken to complete the trip at each speed.
The speed of the truck affects the time taken for the trip. Divide the distance (100 miles) by the speed to get the time taken for the trip at each speed.
Step 4: Calculate the cost of the driver's time at each speed.
The driver is paid $8 per hour. Multiply the time taken for the trip at each speed by the driver's hourly rate to find the cost of the driver's time at each speed.
Step 5: Calculate the total cost of the trip at each speed.
The total cost of the trip at each speed is the sum of the cost of fuel and the cost of the driver's time.
Step 6: Find the speed at which the total cost is minimum.
Compare the total cost at each speed and find the speed at which the total cost is the smallest. This will be the speed at which the truck should drive to minimize the cost of the trip.
Note: It's important to note that while finding the speed that minimizes the cost of the trip, we assume that the truck can sustain any speed between 80 and 110 mph without any additional costs or limitations. In reality, there might be other factors such as fuel efficiency at higher speeds, vehicle wear, and safety considerations that would need to be taken into account.