Beginning Algebra
Tutorial 32: Formulas
Learning Objectives
After completing this tutorial, you should be able to:
 Solve a formula for a given variable.
 Solve problems involving formulas.

Introduction
In this tutorial we will be solving problems using
formulas to help
us. We will be looking at such formulas as area of a
rectangle,
volume of a sphere, Pythagorean theorem and so on. After going
through
this tutorial, you will be an old pro at solving problems involving
formulas. 
Tutorial
A formula is an equation that involves two or more
variables that have
a specific relationship with each other. 
Formulas for Some
2Dimensional Figures

Area of a Parallelogram

In other words, to get the area of a parallelogram,
you multiply the base and height.
Keep in mind that a rectangle and square are two special
types of parallelograms,
and would follow this same formula.
So what would be the area of the following parallelogram
be?


*base = 15 and height = 9
*multiply

The area is 135 square inches. 
Area of a Triangle

In other words, to get the area of a triangle,
you take one half of the base times the height
So what would be the area of the following triangle?


*base = 11 and height = 5
*multiply

The area is 27.5 square units. 
Area and Circumference of a
Circle
Area:
Circumference:

In other words, to get the area of a circle,
you take pi times the radius squared. And to get the
circumference
of a circle, you take 2 times pi times the radius.
So what would be the area and circumference of the
following circle?


*Area
*radius = 8
*8 squared is 64
*Circumference
*radius = 8
*multiply

The area is 64 pi square centimeters.
The circumference is 16 pi centimeters. 
Formulas for Some
3Dimensional Figures

Surface Area and Volume of a
Rectangular Solid
Surface Area:
Volume:

In other words, to get the surface area of a
rectangular solid, you
take two times the length times the width plus two times the length
times
the height plus tow times the width times the height. And to get the
volume
of a rectangular solid, you take the length times the width times the
height.
So what would be the surface area and volume of the
following rectangular
solid?


*Surface Area
*length = 5, width = 2 and height
= 3
*multiply
*Volume
*length = 5, width = 2 and height
= 3
*multiply 
The surface area is 62 square feet.
The volume is 30 cubic feet. 
Surface Area and Volume of a
Sphere
Surface Area:
Volume:

In other words, to get the surface area of a sphere,
you take four
times pi times the radius squared. And to get the volume of a
sphere,
you take the 4/3 of pi times the radius cubed.
So what would be the surface area and volume of the
following sphere?


*Surface Area
*radius = 18
*multiply
*Volume
*radius = 18
*multiply

The surface area is 1296 pi square
units.
The volume is 7776 pi cubic units. 
Surface Area and Volume of a
Right Circular Cylinder
Surface Area:
Volume:

In other words, to get the surface area of a right
circular cylinder,
you take two times pi times the radius times the height and add that to
two times pi times the radius squared. And to get the volume of a
right circular cylinder, you take pi times the radius squared times the
height.
So what would be the surface area and volume of the
following right
circular cylinder?


*Surface Area
*radius = 5 and height = 10
*multiply
*Volume
*radius = 5 and height = 10
*multiply 
The surface area is 150 pi square
millimeters.
The volume is 250 pi cubic millimeters. 
Solving a Formula
for a Specified Variable

Basically, you want to get the variable you are
solving for alone
on one side and everything else on the other side (including variables
you are not solving for) using INVERSE operations.
Even though there is more than one variable in a
formula, you solve
for a specific variable using the
exact same steps that you do with an equation in one variable, as shown
in Tutorial 14: Solving Linear Equations (Putting it all together).
It is really easy to get overwhelmed when there is more
than one variable
involved. Sometimes your head feels like it is spinning when you
see all of those variables. Isn’t math suppose to be about
numbers?
Well, just remember that a variable
represents
a number, so if you need to move it to the other side of the equation
you
use inverse operations, just like you would do with a number. 
Example
1: Solve the equation for L. 
Do you recognize this formula?
This happens to be the formula for the perimeter of a rectangle, where P = perimeter, L = length, and W = width.
In this problem, we need to solve for L.
This means we need to get L on one side
and
EVERYTHING ELSE on the other side using inverse operations.
Let’s solve this formula for L: 

*Inverse of add 2W is sub. 2W
*Inverse of mult. by 2 is div.
by 2
*Formula solved for L

Example
2: Solve the equation
for r. 
Do you recognize this formula?
This happens to be the formula for the circumference
of a circle, where C =
circumference, = pi, and r = radius.
In this problem, we need to solve for r.
This means we need to get r on one side
and
EVERYTHING ELSE on the other side using inverse operations.
Let’s solve this formula for r: 

*Inverse of mult. by 2pi is
div. by 2pi
*Formula solved for r 
Example
3: Solve the equation for y. 
This is an equation for a line.
In this problem, we need to solve for y.
This means we need to get y on one side
and
EVERYTHING ELSE on the other side using inverse operations.
Let’s solve this formula for y: 

*Inverse of add 5x is sub. 5x
*Inverse of mult. by 4 is div.
by 4
*Formula solved for y

Example
4: Solve the equation for h. 
Do you recognize this formula?
This happens to be the formula for the volume
of a rectangular solid, where V = volume, l = length, w = width, and h = height.
In this problem, we need to solve for h.
This means we need to get h on one side
and
EVERYTHING ELSE on the other side using inverse operations.
Let’s solve this formula for h: 

*Inverse of mult. by lw is div. by lw
*Formula solved for h 
Solving Problems Involving
Formulas

Step 1: Identify the type(s)
of figure(s) in
the problem. 
For example, are you working with a circle, cylinder,
square, etc?
Are you working with more than one figure? These are the
questions
you need to answer. 
Step 2: Identify what
formula(s) you need. 
For example, are you looking for the perimeter, area ,
volume, etc.
of the figure(s) you identify in step 1? 
Step 3: Put the problem
together. 
Sometimes the problem is cut and dry and you just
simply plug in to
a formula and go.
Sometimes you need to do a little figuring. You
may need to add,
subtract, or take a fraction of the formula(s) you came up with in step
2. 
Example
5: One bag of fertilizer will cover 500 square
feet
of lawn. Your rectangular lawn is 70 feet by 50 feet. How
many
bags of fertilizer will you need to cover it? 
Step 1: Identify the type(s)
of figure(s) in
the problem.
AND 
Step 2: Identify what
formula(s) you need. 
Make sure that you read the question carefully several
times.
Since we are needing to find the area
of a rectangle,
we can use this formula:
The variables in this formula represent the following:
A = Area of a rectangle
L = length
W = width 
Step 3: Put the problem
together. 
In this problem,
A = ? = this is the variable
we are looking
for
L = 70
W = 50
Plugging the values into the formula we get:

First, find the area of the lawn: 
For every 500 square feet, you need 1 bag of
fertilizer. So, we
need to see how many times 500 sq. feet goes into 3500 sq. feet to find
the number of bags of fertilizer needed. 
FINAL ANSWER:
7 bags of fertilizer.

Example
6: The diameter of a beach ball was found to be
18 inches. What is the volume of this beach ball? 
Step 1: Identify the type(s)
of figure(s) in
the problem.
AND 
Step 2: Identify what
formula(s) you need. 
Make sure that you read the question carefully several
times.
Since we are looking for the volume
of a sphere,
we can use this formula:
The variables in this formula represent the following:
V = volume of a sphere
r = radius 
Step 3: Put the problem
together. 
In this problem,
V = ? = this is the variable
we are looking
for
r = 9 (radius is
half the diameter,
so r = 18/2 = 9)
Plugging the values into the formula we get:

FINAL ANSWER:
The volume of the beach ball is cubic
inches.

Example
7: A ramp 13 feet long is leaning against a
raised
platform which is 5 feet above the ground. What is the distance
from
the ramp’s contact point with the ground and the base of the platform?

Step 1: Identify the type(s)
of figure(s) in
the problem.
AND 
Step 2: Identify what
formula(s) you need. 
Make sure that you read the question carefully several
times.
Since we are looking for the side of a right
triangle, we can
use the Pythagorean formula:
The variables in this formula represent the following:
a and b = legs
of the right triangle
c = hypotenuse of the right
triangle 
Step 3: Put the problem
together. 
In this problem,
a = ? = this is the variable
we are looking
for
b = 5
c = 13
Plugging the values into the formula we get:


*Square 5 and 13
*Subtract 25 from both sides
*What squared gives you 144? 
FINAL ANSWER:
The distance from the ramp’s contact point with the
ground and the
base of the platform is 12 feet. 
Add if You are
Putting Figures Together

Example
8: A cylindrical pedestal for a statue is to
have
a height of 5 feet and a diameter of 2 feet. The pedestal’s base
is to be a rectangular solid that is 9 feet long, 4 feet wide, and 2
feet
thick. What volume of cement is needed to construct the pedestal
and its base? 
Step 1: Identify the type(s)
of figure(s) in
the problem.
AND 
Step 2: Identify what
formula(s) you need. 
Make sure that you read the question carefully several
times.
Since part of the problem is looking for the volume
of a cylinder, we can use the formula :
The variables in this formula represent the following:
=
volume of the
cylinder
r = radius
h = height
Since part of the problem is looking for the volume
of a rectangular solid, we can use also use the formula:
The variables in this formula represent the following:
=
Volume of the
rectangular solid
l = length
w = width
h = height 
Step 3: Put the problem
together. 
In this problem,
V = ? = this is the variable
we are looking
for
r = 1 (radius is
half the diameter,
so r = 2/2 = 1)
h (of cylinder) = 5
l = 9
w = 4
h (of rectangular solid)= 2
If we take the volume of the cylinder and add it to
the volume of
the rectangular solid, then we will have the volume that we are looking
for:
Plugging the values into the formula we get:

FINAL ANSWER:
The volume of the beach ball is cubic inches.

Subtract if You are
Taking Out Parts of a Figure

Example
9: Using the figure shown, find the area in
square
feet of the middle region in the square? 
Step 1: Identify the type(s)
of figure(s) in
the problem.
AND 
Step 2: Identify what
formula(s) you need. 
Make sure that you read the question carefully several
times.
Since part of the problem involves the area of
a square, we can use the formula :
The variables in this formula represent the following:
=
area of square
s = side
Since part of the problem involves the area
of a circle, we can use also use the formula:
The variables in this formula represent the following:
=
area of the four
quarter circle corners (four quarters = 1 whole circle)
r = radius 

Step 3: Put the problem
together. 
In this problem,
A = ? = this is the variable
we are looking
for
s = 20
r = 10
If we take the area of the square and subtract out
the area of the
four quarter circles (whole circle) we will have the area of the middle
region of the given figure above:
Plugging the values into the formula we get:

FINAL ANSWER:
The area of the middle region is square
feet.

Take a Fraction of a Formula if
You
Only Have a Portion of a Figure

Example
10: A dome is hemispherical in shape with a
radius
of 16 meters and is built using 8 equal sections. What formula
would
describe the surface area of each section? 
Step 1: Identify the type(s)
of figure(s) in
the problem.
AND 
Step 2: Identify what
formula(s) you need. 
Make sure that you read the question carefully several
times.
Since we are looking for the surface area of 1/8 of
a hemisphere
(half of a sphere), we can use the formula :
The variables in this formula represent the following:
SA = surface area
r = radius 

Step 3: Put the problem
together. 
In this problem,
SA = ? = this is the
variable we are looking
for
r = 16
Plugging 16 in for r we get:
Simplifying the expression we get: 
FINAL ANSWER:
The surface area of one section is square
meters.

Practice Problems
These are practice problems to help bring you to the
next level.
It will allow you to check and see if you have an understanding of
these
types of problems. Math works just like
anything
else, if you want to get good at it, then you need to practice
it.
Even the best athletes and musicians had help along the way and lots of
practice, practice, practice, to get good at their sport or instrument.
In fact there is no such thing as too much practice. To get the most out of these, you should work the
problem out on
your own and then check your answer by clicking on the link for the
answer/discussion
for that problem. At the link you will find the answer
as well as any steps that went into finding that answer. 
Practice
Problems 1a  1b: Solve each equation for the specified
variable.
Practice
Problems 2a  2c: Solve the following word problems.
2a. Sally is training for the Olympics. She
likes to run
around a circular track that has a diameter of 60 yards, 20 times
during a workout. How many yards does she run during her workout?
(answer/discussion
to 2a) 
2b. A ramp 5 feet long is leaning against a
raised platform which
is 4 feet above the ground. What is the distance from the ramp’s
contact point with the ground and the base of the platform?
(answer/discussion
to 2b) 
2c. In the figure, ABCD is a square, with each side of
length 8 inches.
The width of the border (shaded portion) between the outer square EFGH
and ABCD is 2 inches. Find the area of the border.

Need Extra Help on these Topics?
Last revised on August 3, 2011 by Kim Seward.
All contents copyright (C) 2001  2010, WTAMU and Kim Seward. All rights reserved.

