copyright © 2010 pearson education, inc. all rights reserved sec 2.2 - 1


Linear Equations and Applications

Chapter 2


2.2

Formulas and Percent

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 2.2 - 4

2.2 Formulas and Percent

Objectives

1. Solve a formula for a specified variable.

2. Solve applied problems using formulas.

3. Solve percent problems.

4. Solve problems involving percentincrease or decrease.



A Mathematical Model

Mathematical Model

A mathematical model is an equation or inequality

that describes a real situation. Models for many

applied problems already exist and are called

formulas.



A Formula

FormulaA formula is a mathematical equation in which

variables are used to describe a relationship.



Using formulas to describe a relationship

Relationship Mathematical Formula

Perimeter of a triangle:

a

b

ch

Area of a triangle:



Using variables to describe a relationship

Relationship Mathematical Formulae

h

r

Volume of a cone:

Surface area of a cone:




Relationship Mathematical Formulae

Celsius to Fahrenheit:

Fahrenheit to Celsius:

Celsius Fahrenheit




Relationship Mathematical Formula

Percent Acid, P:

Base

Acid



Solving a Formula for a Specified Variable

Sometimes the formula is solved for a different variable than the one to be found.

One mathematical model tell us that voltage, V, in a circuit is equal to current, I, times resistance, R.

V = I RTo determine the amount of resistance in a

circuit, it would help to first solve the formula for R.




Solve the formula V = IR for R.

We solve this formula for R by treating V and I as constants (having fixed values) and treating R as the only variable. Begin by writing the formula so that the variable for which we are solving, R, is on the left side.I R = V




Finally, we use the properties of the previous section to isolate the variable R.

I R = V

Divide by I.I R = VI I

R =IV




Solve the formula P = a + b + c for b.

We solve this formula for b by treating P, a and c as constants (having fixed values) and treating b as the only variable. Begin by writing the formula so that the variable for which we are solving, b, is on the left side.

a + b + c = P




We now solve for b.

a + b + c = Pa + b + c + (–a) = P + (–a)b + c = P – ab + c + (–c) = P – a + (–c)b = P – a – c

Add. Prop. of Eq.

Add. Prop. of Eq.




B

b

h

This formula gives the relationship between the height, h, and two bases, B and b, of a trapezoid and its area, A.




Mult. Prop. of Equality.

Assoc. Prop.

Inverse Prop.

Identity Prop.




Add. Prop. of Equality.

Divide by h.

Distributive Prop.

Inverse Prop.



Solving Applied Problems Using Formulas

h

b

l

The volume of a triangular cylinder is given by:

If the volume of a triangular cylinder is 880 cm3, the base is 10 cm, and the length is 22 cm, find the height.




Continued.

First, solve the equation for h.




Continued.

Second, find the height, h, by substituting the given values of V, b, and l into this formula:

The height of the triangular cylinder is 8 cm.



Solving Percent Problems

The word “percent” means per 100. For example, 5 percent means 5 per one hundred. Percent is written with the % symbol (e.g., 5%).

Let a represent the partial amount of b, the base, or whole amount. The following formula can be used to solve percent problems.

(repreamount

percentb

sented as a decimaa

l)se

a

b



Solving Percent Problems

A chain saw requires 4 ounces of a certain oil to be mixed with 60 ounces of gasoline. What is the percent of oil in the mixture?

The whole amount of the mixture will be:

Oil 4 ounces

Gas 60 ounces

Total 64 ounces

Let x represent the percent of oil in the mixture. Then the percent of oil in the mixture is:



Interpreting Percents from a Graph

F 8%

A 16%

B 28%

C 36%

D 12%

The pie chart shown below represents the distribution of grades in Analytic Geometry 122 last year. Use the information in the chart to estimate how many B’s will be given in a new class of size 70 students.



Interpreting Percents from a Graph

F 8%

A 16%

B 28%

C 36%

D 12%

According to the chart, 28% of the students should get a grade of B. Let x represent the number of students getting a B.

Thus, about 20 students will get a B.



Solving Problems About Percent Increase or Decrease

A hardware store marked up a water heater from their cost of $600 to a selling price of $708. What was the percent markup?

amount of increasepercent increase =

base708 600

= 600

x

108 =

600x

= 0.18x

The water heater was marked up 18%.

copyright © 2010 pearson education, inc. all rights reserved sec 2.2 - 1

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