SOLUTION: A truck driving 260 miles over a flat interstate at a constant rate of 50 miles per hour gets 7 miles to the gallon. Fuel costs $3.50 per gallon. For each mile per hour increase in

Question 511958: A truck driving 260 miles over a flat interstate at a constant rate of 50 miles per hour gets 7 miles to the gallon. Fuel costs $3.50 per gallon. For each mile per hour increase in speed, the truck loses a tenth of a mile per gallon in its mileage. Drivers get $27.50 per hour in wages and fixed costs for running the truck amount to $11.33 per hour. What constant speed (between 50 mph and the speed limit of 65 mph) should the truck drive to minimize the total cost of the trip?
Part I: To solve a problem like this, it is a good idea to start with calculating an actual example. We will begin by finding the total cost if the truck is driven at 50 miles per hour. The cost is made up of two components: the gas cost and the time cost. Be sure to include the correct units with each value below.
A. Let𠏋 start out by finding how long the trip will take.
The length of time required for the trip is ____________________. (Do not round.)
B. Now, with this time known, how much will it cost to pay the driver and run the truck?

The amount needed to pay the driver and run the truck is 洵洵洵洵洵洵洵____________________. (Round to nearest cent.)
C. Next determine, at 7 miles per hour for 260 miles, how much gas will be required.

The amount of gas required is ______________柯______. (Do not round. Leave as proper fraction.)
D. With the amount of gas known, how much will the gas cost to make the 260 miles?

The cost of the gas is 洵洵洵洵洵洵洵_____________________. (Round to nearest cent.)
E. Finally we can find the TOTAL cost.
The total cost for the trip is 洵洵洵洵洵洵洵_____________________. (Round to nearest cent.)
Part II: The preceding process should have illuminated the basic procedure we will use to find the total cost of a trip. Next we will find the total cost if the truck is driven at 65 miles per hour. As in Part I, include the correct units with each value.
A. Let𠏋 find how long the trip will take.
The length of time required for the trip is ____________________. (Do not round.)
B. Now, with this time known, how much will it cost to pay the driver and run the truck?

The amount needed to pay the driver and run the truck is 洵洵洵洵洵洵洵____________________. (Round to nearest cent.)
C. Next, to begin determining the gas cost, we need to find the mileage (miles per gallon of gas) when the truck is travelling at 65 miles per hour.

The mileage at 65 miles per hour is ______________柯______. (Do not round.)
D. With the gas mileage known, how much gas will be needed for the 260 miles?

The amount of gas required is 洵洵洵洵洵洵洵_____________________. (Do not round. Leave as proper fraction.)
E. With the amount of gas known, how much will the gas cost to make the 260 miles?

The cost of the gas is 洵洵洵洵洵洵洵_____________________. (Round to nearest cent.)
F. Finally we can find the TOTAL cost.
The total cost for the trip is 洵洵洵洵洵洵洵_____________________. (Round to nearest cent.)
Part III. We should now have a good process for determining the total cost of a trip for any rate of speed greater than or equal to 50 miles per hour. Next is to create a Total Cost function using X as the unknown rate in miles per hour. Simplify your answers and remember to include units. As you work through each step, test your answers by plugging in 50 mph and then 65 mph and comparing with results from parts I and II.
A. Let𠏋 find how long the trip will take.
The length of time required for the trip is _____________________.
B. Now with this time known, how much will it cost to pay the driver and run the truck?

The amount of money needed to pay the driver is 洵洵洵洵洵洵洵_____________________.
C. Next, to begin determining the gas cost, we need to find the mileage (miles per gallon of gas) when the truck is travelling at X miles per hour.

The mileage at X miles per hour is ______________柯______.
D. With the gas mileage known, how much gas will be needed for the 260 miles?

The amount of gas required is 洵洵洵洵洵洵洵_____________________.
E. With the amount of gas known, how much will the gas cost to make the 260 miles?

The cost of the gas is 洵洵洵洵洵洵洵_____________________.
F. Now we can find the TOTAL cost function. Express your function as C(X) =
TOTAL Trip Cost Function is 洵洵洵洵洵洵洵___________________________________
G. The last thing we should do is verify that this is the correct function by evaluating it at 50 mph and 65 mph to see if we get the same values we have previously computed.

F(50) = ____________________________
F(65) = ____________________________
Answer by richard1234(7193) (Show Source): You can put this solution on YOUR website!
Please do not post a 30-part problem into one question. Tutors are unlikely to answer every one, unless you hire a paid tutor.

I'll give you some tips for this huge problem. Really pay attention on what each problem is asking for. Look for patterns regarding the cost due to wages and gas, versus the speed. Once you notice a pattern, try to generalize and create a function for cost in terms of speed. To minimize the cost, simply find the minimal value of the function (this is easy for certain functions, may require calculus on others).
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