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2 Most of the word problems you do in math classes are not actually related to real life. Textbooks try to pretend they are by using real life data, but they do not use that data in the same way that it is actually used. However, this section is different. We will finally take a look at some calculations that are actually done in real life situations to make actual decisions.

3 The first type of analysis is known as a break-even analysis. The textbook uses the business example of trying to compute when you turn a profit, but this type of analysis can also be used when comparing loan types and making insurance decisions. A break-even analysis begins with the following formula: (Profit) = (Revenue) (Cost) The word cost refers to the cost of manufacturing the product and the word revenue refers to the amount of money that crosses the table when you sell it. Example: If it costs \$5 to create a widget and you sell it for \$7, you make a profit of \$2.

4 There are two types of costs: fixed costs and variable costs. The fixed costs are those things that you would have to pay regardless of the number of widgets you produce. Examples of these costs would be rent, tools, and insurance. The variable costs are the costs that are associated to actually producing an individual widget. For example, materials and labor for making your widgets are variable because if you make a large number of widgets you will need a lot of materials and a lot of workers to build them (assuming you couldn t buy a machine to do it).

5 Before we can get to solving problems, we need to introduce one new mathematical idea. A function is an instruction for how to compute a value. Example: The function that computes the sale price of gas at \$4 per gallon is f(x) = 4x. The f(x) is read f of x and means that you can give it an x value and it will tell you how to compute the sale price of gas. Example: Compute the sale price of 5 gallons of gas. f(x) = 4x f(5) = 4(5) = 20 Notice that all we did was replace the variable x with the specific number we were interested in using. We could have used any number we wanted.

6 We can use whatever letters we want (or even a whole word) as the name of the function and the variable. Example: The function that computes the sale price of gas at \$4 per gallon is S(g) = 4g. The choice of the letter S for Sale price and the letter g for number of Gallons make sense. In math, you will find a tendency towards using the letter x for generic variables and the letters f and g for generic functions.

7 Example: A furniture company is considering creating a new line of chairs. The fixed costs will be \$90,000, and it will cost \$25 to produce each chair. They intend to sell the chairs for \$48 each. Find the total cost of producing 1000 chairs. Cost of 1000 producing chairs = = = Find the total cost of producing chairs. Cost of producing chairs = = = Find the total cost C(x) of producing x chairs. C(x) = x

8 Example: A furniture company is considering creating a new line of chairs. The fixed costs will be \$90,000, and it will cost \$25 to produce each chair. They intend to sell the chairs for \$48 each. Find the revenue of selling 1000 chairs. Revenue of selling 1000 chairs = = Find the revenue of selling chairs. Revenue of selling chairs = = Find the revenue R(x) of selling x chairs. R(x) = 48x

9 Example: A furniture company is considering creating a new line of chairs. The fixed costs will be \$90,000, and it will cost \$25 to produce each chair. They intend to sell the chairs for \$48 each. Number of chairs x Revenue x Cost x Since we have the cost and the revenue, we can compute the profit for selling each number of chairs using the profit formula: (Profit) = (Revenue) (Cost)

10 Example: A furniture company is considering creating a new line of chairs. The fixed costs will be \$90,000, and it will cost \$25 to produce each chair. They intend to sell the chairs for \$48 each. Number of chairs x Revenue x Cost x Profit x What do the numbers mean? If the company was only able to sell 1000 chairs, then it would lose \$67,000. If the company was able to sell 10,000 chairs, then it would make a profit of \$140,000 This leads us to an interesting question: How many chairs must be sold in order to break-even?

11 Example: A furniture company is considering creating a new line of chairs. The fixed costs will be \$90,000, and it will cost \$25 to produce each chair. They intend to sell the chairs for \$48 each. Number of chairs x Revenue x Cost x Profit x Find the break-even point. 23x = 0 23x = x The break-even point is about 3913 chairs.

12 This type of analysis can be used in many circumstances. Should I refinance my mortgage at a lower interest rate? There are some up front costs when you refinance a mortgage, and it s important to determine whether you will actually save money over the life of the mortgage in order to justify paying the fees. What deductible should I have on my insurance policy? If you have a small deductible, you pay more every month, but you pay less when you make a claim. If you have a large deductible, you pay less every month, but you pay more when you make a claim. If you think you can go long enough without making a claim to break even, then you should take the larger deductible because the money you saved in premiums can be used to pay the extra costs.

13 Another type of analysis comes from economics, specifically the laws of supply and demand. There are two basic principles that we will need. If the price increases, demand will decrease. Similarly, if the price decreases, demand will increase. (If something costs more, fewer people will want to buy it.) If the price increases, the supply will increase. Also, if the price decreases, the supply will decrease. (If something sells for a higher price, companies will want to produce more of it to make more money.) We can represent these two principles by using a graph.

14 Quantity Quantity Supply Demand Price When we merge these two pictures together, we get an interesting phenomenon. Price

15 Quantity Supply Equilibrium quantity Equilibrium point Equilibrium price Demand Price The equilibrium price in some sense is the right price to sell things because the market will buy everything that the companies produce. The equilibrium quantity is the number of the product that needs to be produced.

16 While it is helpful to understand the concepts, for the problems that will be worked on in this class, you will only need to know the mechanics. Set the demand function equal to the supply function. D(p) = S(p) Solve for p to find the equilibrium price. Plug this value back into either the demand or supply function to get the equilibrium quantity. The equilibrium point is (equilibrium price, equilibrium quantity).

17 Example: Find the equilibrium price for the demand and supply functions D(p) = p S(p) = p D(p) = S(p) p = 200+4p 64p = 800 p = 12.5 S(p) = 200+4p S(12.5) = 200+4(12.5) = = 250 The equilibrium point is (12.5, 250).

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