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

​Modeling Systems of Equations: Making Money (and losing it)

​Applied Math

​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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Multiple Choice

Which of the following is the best definition of REVENUE?

• The total amount of income a business makes from selling products or services

• The amount of money a business has LEFT over after paying for their costs.

• The total amount of money a business spends.

• The amount of money a business loses every month.

Multiple Choice

Which of the following is the best definition of COSTS?

• The total amount of income a business makes from selling products or services

• The amount of money a business has left over after paying for materials.

• The total amount of money a business spends.

​Break Even Point

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.​

​Example

​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.

​Example

​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

​Example

​ 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.​

​Example - Graph

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

R(2500) = 600(2500) = 1,500,000​

Multiple Choice

When a business has made enough money to pay its costs and begin to make a profit, it has reached its

• break-even point

• variable-cost margin

• fixed cost

• selling price

Multiple Choice

The formula for total cost =

• Fixed cost x variable cost

• Fixed cost - variable cost

• Variable cost - fixed cost

• Fixed cost + variable cost

Multiple Choice

If Mr. Edwards had a fixed cost of $500 and a variable cost of $5 per chair and he produced 20 chairs. His total cost would be

• $500

• $550

• $600

• $505

​The Profit Function

​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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Multiple Choice

Is there a difference between the profit function and the break even point?

• Yes

• No

Multiple Choice

How many radios must be produced and sold for the company to break even?

• 200

• 300

• 500

• 600

Multiple Choice

How much money will the company need to make in order to break even?

• 10,000

• 15,000

• 20,000

• 25,000

​Your Turn / Exit Ticket

​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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