**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.

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?

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?

**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?

***Circumference**

***radius = 8**

***multiply**

The circumference is 16

**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?

***Volume**

***length = 5, width = 2 and height
= 3 **

***multiply**

**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?

***Volume**

***radius = 18**

***multiply**

**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?

***Volume**

***radius = 5 and height = 10**

***multiply**

**The volume is 250 pi cubic millimeters.**

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.**

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.

This happens to be the formula for the perimeter of a rectangle, where

**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 mult. by 2 is div.
by 2**

***Formula solved for L**

Do you recognize this formula?

This happens to be the formula for the**circumference
of a circle**, where *C *=
circumference, = pi, and *r *= radius.

This happens to be the formula for the

**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**

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 mult. by 4 is div.
by 4**

***Formula solved for y**

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.

This happens to be the formula for the

**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*:

***Formula solved for h**

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.

For example, are you looking for the perimeter, area ,
volume, etc.
of the figure(s) you identify in step 1?

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.

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

In this problem,

*A *= ? = this is the variable
we are looking
for

*L = *70

*W = *50

**Plugging the values into the formula we get:**

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.**

**7 bags of fertilizer.**

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

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:**

***Multiply**

**The volume of the beach ball is cubic
inches.**

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

In this problem,

*a *= ? = this is the variable
we are looking
for

*b *= 5

*c *= 13

**Plugging the values into the formula we get:**

***Subtract 25 from both sides**

***What squared gives you 144?**

**The distance from the ramp’s contact point with the
ground and the
base of the platform is 12 feet.**

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

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:**

**The volume of the beach ball is cubic inches.**

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

In this problem,

*A *= ? = this is the variable
we are looking
for

s = 20

*r *= 10

s = 20

**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:**

**The area of the middle region is square
feet.**

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

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: **

**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)

(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)

(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?**

**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.**

Last revised on August 3, 2011 by Kim Seward.

All contents copyright (C) 2001 - 2010, WTAMU and Kim Seward. All rights reserved.