Using Linear Formulas by  Example 

-- Part 1.2 --

Most of these problems are easy enough to solve without using formulas. Our goal here is to use simple examples to learn how to express problems mathematically. Express each example with a formula or equation.

Find the total cost of producing x units of widgets. The fixed costs are $2,000 and the cost per unit is $50. How much would it cost to produce 1,000 units? 5,000 units?

Quantities

Step

Use the Formula

  • Fixed Costs = 2,000

  • Cost per Unit = 50

  • Let x = Number of Units

  • Let C = Total Cost

C = 50x + 2,000

Identify the known quantities.

Identify the unknown quantities.

Create a model (or equation)

For 1,000 units, let x = 1,000.
C = 50(1,000) + 2,000 
C = 50,000 + 2,000 = 52,000

For 5,000 units, let x = 5,000.
C = 50(5,000) + 2,000 
C = 250,000 + 2,000 = 252,000

Interpret answers: It costs $52,000 to produce 1,000 widgets and $252,000 to produce 5,000 widgets. Are the answers reasonable?

Find the total cost of producing x units of radios. The fixed costs are $500 and the cost per unit is $20. How much would it cost to produce 1,000 units? 5,000 units?

[Solution]

A bookstore pays $60 for a math textbook. The bookstore adds 33% to the cost of each book. How much revenue would the bookstore make if it sold 100 books? 1,000 books?

Quantities

Step

Use the Formula

Bookstore Cost per Unit = 60

Mark up per Unit = .33 × 60 = 19.80
Student Price = 60 + 19.80 = 79.80

Let x = Number of Units
Let R = Revenue R = 79.80x 

Identify the known quantities.

Identify the unknown quantities.

Create a model (or equation)

For 100 units, let x = 100.
R = 79.80(100) =  $7,980

For 1,000 units, 
let x = 1,000.
R = 79.80(1,000) =  $79,800

A school wants to sell calculators for a fundraiser. The school pays $25 dollars for each calculator and they can sell them for 40% more than the purchase price. How much revenue would the school make if they sold 50 calculators? 350 calculators?

[Solution]

From example 2: A bookstore pays $60 for a math textbook. The bookstore adds 33% to the cost of each book. How much profit would the bookstore make if it sold 100 books? 1,000 books?

Quantities

Step

Use the Formula

There are many costs a business incurs such as taxes, rent, employees,  insurance and more taxes. We will ignore those costs in this problem.
Revenue R = 79.80x
Cost C = 60x
Let x = Number of Books
Let P = Profit
P = R - C = 79.80x - 60x

P = 19.80x

Identify the known quantities
Identify the unknown quantities· 

Profit is Revenue minus Costs

Create a model (or equation)

For 100 units, let x = 100.
R = 19.80(100) =  $1,980

For 1,000 units, 
let x = 1,000.
R = 19.80(1,000) =  $19,800

 A company's cost is represented by C = 120x +10,000 where x is the number of units sold. The company sells each unit for $300 so revenue is R = 300x. How much did the company make if they sell 50 units. 100 units?

[Solution]

Perimeter of rectangle   Solve for w

Formula   P = 2l + 2w

Step

 P = 2l + 2w

P - 2l = 2w

Original Equation

Subtract 2l from both sides


Divide by 2

 Perimeter of rectangle   Solve for l

                               Formula   P = 2l + 2w

[Solution]

Simple Interest formula where A = amount, P = principal
        r = interest rate per period, t = time.
   
Formula  A = P + Prt

Step

 A = P + Prt

 A = P(1 + rt)

Original Equation

Factor out P


Divide by
1 + rt

 

Done

   Solve for r.   Looks tricky but it is not!

                              

[Solution]

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joe mcdonald

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Last Update 09.23.2010