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Copyright © 2010 Pearson Education, Inc. All rights reserved Sec 2.2 - 2
Linear Equations and Applications
Chapter 2
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.
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 2.2 - 5
2.2 Formulas and Percent
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.
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 2.2 - 6
2.2 Formulas and Percent
A Formula
FormulaA formula is a mathematical equation in which
variables are used to describe a relationship.
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 2.2 - 7
2.2 Formulas and Percent
Using formulas to describe a relationship
Relationship Mathematical Formula
Perimeter of a triangle:
a
b
ch
Area of a triangle:
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 2.2 - 8
2.2 Formulas and Percent
Using variables to describe a relationship
Relationship Mathematical Formulae
h
r
Volume of a cone:
Surface area of a cone:
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 2.2 - 9
2.2 Formulas and Percent
Using variables to describe a relationship
Relationship Mathematical Formulae
Celsius to Fahrenheit:
Fahrenheit to Celsius:
Celsius Fahrenheit
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 2.2 - 10
2.2 Formulas and Percent
Using variables to describe a relationship
Relationship Mathematical Formula
Percent Acid, P:
Base
Acid
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 2.2 - 11
2.2 Formulas and Percent
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.
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 2.2 - 12
2.2 Formulas and Percent
Solving a Formula for a Specified Variable
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
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 2.2 - 13
2.2 Formulas and Percent
Solving a Formula for a Specified Variable
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
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 2.2 - 14
2.2 Formulas and Percent
Solving a Formula for a Specified Variable
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
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 2.2 - 15
2.2 Formulas and Percent
Solving a Formula for a Specified Variable
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.
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 2.2 - 16
2.2 Formulas and Percent
Solving a Formula for a Specified Variable
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.
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 2.2 - 17
2.2 Formulas and Percent
Solving a Formula for a Specified Variable
Mult. Prop. of Equality.
Assoc. Prop.
Inverse Prop.
Identity Prop.
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 2.2 - 18
2.2 Formulas and Percent
Solving a Formula for a Specified Variable
Add. Prop. of Equality.
Divide by h.
Distributive Prop.
Inverse Prop.
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 2.2 - 19
2.2 Formulas and Percent
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.
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 2.2 - 20
2.2 Formulas and Percent
Solving Applied Problems Using Formulas
Continued.
First, solve the equation for h.
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 2.2 - 21
2.2 Formulas and Percent
Solving Applied Problems Using Formulas
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.
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 2.2 - 22
2.2 Formulas and Percent
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
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 2.2 - 23
2.2 Formulas and Percent
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:
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 2.2 - 24
2.2 Formulas and Percent
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.
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 2.2 - 25
2.2 Formulas and Percent
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.
Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 2.2 - 26
2.2 Formulas and Percent
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%.