Trouble with Maximum Minimum

[bobbyjay]

The Question:
Assume that the operating cost of a certain truck (excluding driver's wages) is \(\displaystyle 12 +\frac{x}{6}\) cents per mile when the truck travels at x miles per hour. If the driver earns $6 per hour, what is the most economical speed to operate the truck on a 400 mile turnpike where the minimum speed limit is 40mph and the maximum speed limit is 70mph?
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I'm a little confused on how to even begin this problem. From what I'm reading, the truck driver earns $6/hr. However, while he is driving, it costs him \(\displaystyle 12 +\frac{x}{6}\) cents per mile when the truck travels x miles per hour? So the faster he drives, the more it'll cost him. So the question is, what is best speed to earn the most?

If so, how do I even set this one up?

Thanks!

 

[mmm4444bot]

The phrase "most economical" denotes lowest total operating-cost.

You need to determine a function that outputs the total operating-cost in terms of x. Differentiating that function will allow you to find the value of x that optimizes the total operating-cost (i.e., the speed that results in lowest total operating-cost).

The total operating-cost is the sum of two things: (1) the number of cents that it costs to operate the truck for 400 miles at speed x, and (2) the dollars paid to the driver for the number of hours driven along that 400 miles of road.

You know how to write an expression for the number of hours driven in terms of x, yes? (Think: distance equals rate times time.)

Does my post help you to get started?

If you need more assistance, please ask specific questions.

Cheers :cool:

 

[bobbyjay]

I'm not exactly sure on how to set up the function problem. I just don't see how to piece it all together.

Since the operating costs is 12 + x/6 . I would think that it is

f(x) = 12 + (x/6) + 6x

The first half (12 + x/6) comes from the equation given. But I don't think the second half is correct. The truck driver gets $6 an hour. Therefore $6 multiplied by x number of hours. But then the x is in terms of hours, which would not work with the first given half (I underlined the first half, so you know what I'm talking about).

And just for a quick refresher, finding the equation to this problem gives me the total costs. However, why must I add them together? Won't I have to subtract what the truck driver earns from the operating costs to get the total amount of money he made?

Isn't the derivative the slope of a line at any given point? So then how would finding the derivative to this problem help me with the answer?

 

[galactus]

Be careful, the cost per miles is given in cents, while the driver gets 6 dollars per hour.

Thus, assuming we use dollars, the cost function would be \(\displaystyle \frac{12+x/6}{100}=\frac{x+72}{600}\)

It is a 400 miles trip, so the operating cost is \(\displaystyle 400\cdot \frac{x+72}{600}=\frac{2(x+72)}{3}\)

For the driver, distance = rate times time.

\(\displaystyle xt=400\)

\(\displaystyle t=\frac{400}{x}\)

Multiply by 6 to get the driver's total pay.

\(\displaystyle 6\cdot \frac{400}{x}=\frac{2400}{x}\)

Add these two functions together, then differentiate, set to 0 and solve for x. x being the optimum speed.

It should be between 40 and 70.

 

[mmm4444bot]

[Will] I have to subtract what the truck driver earns from the operating costs to get the total amount of money he made?

No.

What the driver earns IS the total amount of money the driver makes.

The driver neither owns the truck nor operates the trucking company. The driver's wages are a cost to the company. Hence, the driver's wages must be added to the expense of operating the truck, in order to calculate the total-operating cost to the company.



[Is] the derivative the slope of a [tangent] line [to the curve of a function] at any given point? Yes.


So then how would finding the derivative to this problem help me with the answer?

The curve of the total-operating-cost function has a lowest value at its minimum; hence, the slope of the tangent line at that point on the curve is ZERO.

Once you have the derivative function, you find the value of x where this slope is 0.

​Is this exercise the very first exposure that you've had concerning optimizations using a derivative?

.