Beginning Algebra
Tutorial 32: Formulas

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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

Formulas

A formula is an equation that involves two or more variables that have a specific relationship with each other.

Formulas for Some
2-Dimensional Figures

Area of a Parallelogram

parallel lines

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?

parallel lines

*base = 15 and height = 9
*multiply

The area is 135 square inches.

Area of a Triangle

triangle

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?

triangle

*base = 11 and height = 5
*multiply

The area is 27.5 square units.

Area and Circumference of a Circle

circle

Area:
area of a circle

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?

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
3-Dimensional Figures

Surface Area and Volume of a Rectangular Solid

rectangle

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?

rectangle

*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

sphere

Surface Area:
surface area sphere

Volume:
volume shpere

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?

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

cylinder

Surface Area:
surface area cylinder

Volume:
volume cylinder

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?

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.

Area of a Rectangle

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:

*Multiply

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.

*Divide

FINAL ANSWER:

7 bags of fertilizer.

Volume of a Sphere

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:

*Cube 9

*Multiply

FINAL ANSWER:

The volume of the beach ball is cubic inches.

Pythagorean Theorem

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?

example 7c

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:

example 7d

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:

*Multiply

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?

example 9a

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:

*Square 20 and 10

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:

*Multiply

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.

1a.

; for T
(answer/discussion to 1a)

1b.

; for y
(answer/discussion to 1b)

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.

problme 2a

(answer/discussion to 2c)

Need Extra Help on these Topics?

The following are webpages that can assist you in the topics that were covered on this page:

http://www.purplemath.com/modules/solvelit.htm
This webpage helps you with solving formulas for a specified variable.

http://www.purplemath.com/modules/perimetr.htm
This webpage involves a combination of problem solving ideas which includes problem solving using formulas.

Go to Get Help Outside the Classroom found in Tutorial 1: How to Succeed in a Math Class for some more suggestions.

WTAMU > Virtual Math Lab > Beginning Algebra