Question

NO LINKS!!! THIS IS NOT AN ASSESSMENT OR TEST! NOT A MULTIPLE CHOICE!!

1. Factory in Wichita's factory is graphed as a piecewise function over the domain [0, 35,000].

a. Write one or two sentences to describe the cost function for the Wichita factory.

b. Write the piecewise function for the cost per units in the Wichita factory. State the function C(x), where x is the number of units produced.

c. What is the cost per unit for 35,000 units?

d. According to the graph, if only one unit was produced, the cost per unit would be $1.00. In reality, the company would not call in its worker and start up the equipment to produce one unit. What safeguard do you think that many manufacturers use to avoid this type of problem?

e. Of the following three statements, choose the best option and write a viable argument showing how it is reflected by the Cost Function for Wichita's factory.
____ The electricity contract with the utility company is structured so that higher daily energy usage is charged at a lower rate.
____ The plant's production processes are performed primarily by robots that are able to work longer hours, when needed, at no additional cost.
_____Overtime wages were required to produce at levels above 25,000 units. ​

Answer 1

Part (a)

We have two separate constant cost functions that's all rolled into one.

If $0 \le x \le 25,000$, then we go with the horizontal line on the left. The cost is constant at 1 dollar per unit.

If instead $25,000 < x \le 35,000$, then the cost is $0.80 or 80 cents per unit, which is visually shown by the horizontal line on the right.

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Part (b)

The answer is

$C(x) = \begin{cases}1.00 \ \text{ if } \ \ x \le 25,000\\0.80 \ \text{ if } \ \ 25,000 < x \le 35,000\\\end{cases}$

this is basically the same as saying C(x) = 1.00 if x is 25,000 or smaller, OR C(x) = 0.80 if x is between 25,000 and 35,000 where we exclude the left endpoint but include the right endpoint.

In short, C(x) has a split identity. It's either equal to 1.00 or it's equal to 0.80 depending on what x is.

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Part (c)

When x = 35000, we're going to focus on the second piece of the piecewise function. Since the cost is constant here, the cost is 80 cents per unit.

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Part (d)

The company would most likely put a rule in that there has to be some minimum. For instance, they might say that the minimum is 10,000 units and the factory cannot produce anything less than this. This is to ensure that they are able to make money since firing up all those machines and paying the workers costs a lot of money.

The managers would have a pretty good idea of where the break-even point is (either past company data, other consulting data, or theoretical values), and that break-even point is most likely where the production minimums are set. Of course, the managers may encourage workers to exceed the production minimums so that the profit isn't small.

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Part (e)

I'm not entirely sure about this one, but I think the statement "The electricity contract with the utility company is structured so that higher daily energy usage is charged at a lower rate" makes the most sense in this case.

The electric company likely charges a flat rate for energy used. The more energy used would lead to a bulk discount of sorts, and that would lead to a lower rate once you exceed 25,000 units produced. The cost per kilowatt is lower, but overall the electric company is pulling in more money from the factory.