Modeling with Systems of Equations Making Money (and Loosing it)

​Revenue and Cost Function

​A company produces and sells x units of a product. The Revenue is the money generated by selling x units of the product.

It's cost is the cost of producing x units of the product.

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R(x) = ( Price per unit sold) x

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C(x) = fixed cost + (cost per unit produced) x ​

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The Break Even Point is the Intersection of the graphs of the revenue and cost functions.​

Finding the Break Even Point.

a. write the cost function

b. write the revenue function

c. Solve as a system of equations.​

​Technology is now promising to bring light, fast, and beautiful wheelchairs to millions of disabled people. A company is planning to manufacture these radically different wheelchairs. Fixed cost will be $500,000 and it will cost $400 to produce each wheelchair. Each wheelchair will be sold for $600.

​a. The cost function is the sum of the fixed cost and variable cost.

C(x) = $500,000 + 400x

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b.​ The Revenue function is the money generated from the sale of x wheelchairs.

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R(x) = 600x

​c. Set up as system of equations

y = 500,000 + 400x

y = ​600x

​ 600x = 500,000 + 400x

-400x = 500,000 - 400x (subtract to get x on

the other side)​

200x = 500,000 (divide by 200)

​200 200

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x = 2500.​

​We have x = 2500 now we have to find the y by Back Substituting

R(2500) = 600(2500) = 1,500,000​

​The profit P(x) generated after producing and selling units of a product is called the Profit Function.

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P(x) = R(x) - C(x)​

​Wheelchair example: P(x) = R(x) - C(x)

P(x) = 600x - (500,000 + 400x)

= 200x - 500,000

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So, at $2500 the company will ​begin making profit

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​Chose from the following scenarios and write and solve a Revenue, Cost, and Profit function and then create a graph demonstrating the break even point.

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​ A company that manufactures small canoes has a fixed cost of $18,000. It costs $20 to produce each canoe. The selling price per canoe is $80. Let x represent the number of canoes produced and sold.

​ A company that manufactures bicycles has a fixed cost of $100,000. It costs $100 to produce each bicycle. The selling price is $300 per bike. Let x represent the number of bicycles produced and sold.

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​Maria is starting a small business selling flowers. She has to pay $150 for a vender’s permit. Each bouquet costs her $7 and she sells them for $15. How many bouquets must Maria sell and how much will she need to earn break even?

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