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High School Precalculus: Help and Review32 chapters | 297 lessons

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Lesson Transcript

Instructor:
*Joshua White*

Josh has worked as a high school math teacher for seven years and has undergraduate degrees in Applied Mathematics (BS) & Economics/Physics (BA).

This lesson will explore the total cost function and the concepts of average and marginal cost. It will also cover how to find the minimum and maximum cost.

How do companies determine the price that they charge to sell certain goods? Although it might seem random, companies frequently use a cost function to determine how many units of an item they should produce and what price they should sell it for. The **cost function** is just a mathematical formula that gives the total cost to produce a certain number of units. Let's take a more in depth look at the cost function and see how it works.

The cost function, usually denoted *C(x)* where *x* represents a positive number and is generally an integer. If you want to know the cost of producing 50 units of an item, you would plug in 50 for every *x* in the cost function, and then, using order of operations, simplify the expression to a number, or dollar, figure.

Besides the total cost, you can use the cost function to find the average cost and marginal cost of production. To find the average cost, you will simply divide the total cost by the total number of units produced. The marginal, or additional, cost represents the cost of producing one additional unit of the good. If you produce 100 battery chargers, the marginal cost will tell you how much extra it costs to produce that 100th charger. To find the marginal cost, you will find the total cost for the unit and subtract from it the total cost for producing one fewer units.

Now let's see how you would actually use the function.

1. The cost function to produce *x* tires is given as *C(x)*=.012*x* + 5,000.

First, let's find the cost to produce 1500 tires. To find this, you can simply plug in 1500 for *x* and then evaluate the cost function:

*C*(1500) = .012*1,500 + 5,000 = $5,018

Thus, it costs $5,018 to produce 1,500 tires.

Now, let's find the average cost of producing those 1500 tires. To find this, simply divide the total cost, $5,018, by the number of tires, 1500. You should get approximately $3.35. Note that although it costs on average $3.35 to produce each tire, the individual cost of producing each tire, or the marginal cost, is not $3.35. Let's see why.

To find the marginal cost of producing the 1500th tire, we can take the total cost of producing 1500 tires and subtract from that the total cost of producing 1499 tires.

*C*(1499) = (.012*1499) + 5000 = $5017.988

If you plug in 1499 for *x* in our original equation, you should get $5,017.99.

$5,018 - $5017.99 = $.01

Subtracting these two values gives $.01 or 1 cent. Thus, the marginal cost of producing the 1500th tire is approximately one cent.

2. The cost function for a property management company is given as *C(x)* = 50*x* + 100,000/*x* + 20,000 where *x* represents the number of properties being managed.

First, let's find the cost of managing 500 properties. Just substitute 500 in for *x* into the formula to find the answer:

*C*(500) = (50*500) + (100,000/500) + 20,000 = 25,000 + 200 + 20,000 = $45,200

Next, let's find the point where the total cost is minimized. Hopefully, you remember that the minimum, or maximum, of any function can be found by taking its derivative and setting that equal to zero to find the critical points. Then, you'll examine the critical points to see if each is a minimum or a maximum or neither.

To find *C'(x)*, you will have to use the power rule, and it may be helpful to rewrite *C(x)* as *C(x)* = 50*x* + 100,000*x*^-1 + 20,000. For *C(x)* you should get:

50 - 100,000/*x*^2

Setting it equal to zero gives you:

0 = 50 - 100,000/*x*^2

100,000/*x*^2 = 50

100,000 = 50*x*^2

2,000 = *x*^2 or *x* 44.721

Now we need to see if this value is a minimum by finding the value of *C(x)* on either side of it. This will show how the slope of the original function changes around the critical point, if at all. For example, if you plug in 44, just to the left of 44.7, to *C(x)*, you should get -1.653.

*C*(44)= 50 - 100,000/(44^2) = 50 - 100,000/(1936) = 50 - 51.6529 = -1.653

If you plug in 45, just to the right of 44.7, to *C(x)*, you should get .617. Since the slope of the cost function, changes from negative (-1.653) to positive (.617), we can confirm that 44.7 is indeed a minimum. The calculations are shown below.

*C*(44)= 50 - 100,000/(45^2) = 50 - 100,000/(2025) = 50 - 49.3827 = .617

However, since you generally can't manage .7 of a property, you would now have to compare 44 and 45 to see which value yields a lower cost by finding the total cost of each. If you do this you should find:

*C*(44) = (50*44) + (100,000/44) + 20,000 = 2200 + 2272.73 + 20,000 = $24,472.73

*C*(45) = (50*45) + (100,000/45) + 20,000 = 2250 + 2222.22 + 20,000 = $24,472.22

Thus, managing 45 units minimizes the overall total cost.

The **cost function** can allow you to determine many different things about a good or service. You can determine how much it will cost to produce a certain number of items. You can also determine the average cost to produce all of those items and the marginal cost of producing a specific item. Finally, you can find how many items will minimize or maximize the total production cost.

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High School Precalculus: Help and Review32 chapters | 297 lessons

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