Linear Formulas

Using Linear Formulas

-- Section 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
Identify the known quantities.	Create a model (or equation)	For 1,000 units, let x = 1,000.
Fixed Costs = 2,000	C = 50x + 2,000	C = 50(1,000) + 2,000
Cost per Unit = 50		C = 50,000 + 2,000 = 52,000
Let x = Number of Units		For 5,000 units, let x = 5,000
Let C = Total Cost		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]