Question

1. The cost per unit for Wichita’s factory is graphed as a piecewise function over the domain [0, 35000].

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

b) Write the piecewise function for the cost per unit for production of units in the Wichita factory. That is, state the function (), where 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

a) The cost function for the Wichita factory is a piecewise function, meaning it consists of different equations or formulas for different intervals or ranges of units produced.

b) Let's denote the cost per unit as C(u), where u is the number of units produced. The piecewise function for the cost per unit for production in the Wichita factory can be represented as follows:

C(u) =

$1.00, for 0 ≤ u < 1

$50.00, for 1 ≤ u < 5000

$40.00, for 5000 ≤ u < 10000

$35.00, for 10000 ≤ u < 25000

$30.00, for 25000 ≤ u ≤ 35000

c) The cost per unit for 35,000 units can be found by evaluating the cost function C(u) at u = 35,000. Based on the piecewise function mentioned above, the cost per unit for 35,000 units would be $30.00.

d) To avoid the problem of calling in workers and starting up the equipment to produce a single unit, many manufacturers implement a concept called "economies of scale." This means that they try to produce units in larger quantities to spread the fixed costs (such as labor, equipment setup, and overhead) over a greater number of units, reducing the cost per unit. By producing more units, the cost per unit decreases, making it more economically viable.

e) The best option that reflects the Cost Function for Wichita's factory is:

The plant's production processes are performed primarily by robots that are able to work longer hours, when needed, at no additional cost.

Argument: The fact that the cost per unit decreases as the number of units produced increases indicates that the production processes are efficient and can handle longer working hours without incurring additional costs. This suggests the presence of automated processes performed by robots, which can work continuously without requiring overtime wages or incurring extra expenses.

Explanation: