# 3.12: Formulas for Problem Solving

Difficulty Level: Basic Created by: CK-12
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What is the formula for the area of a circle? What is the formula for the volume of a sphere? What is the formula for the area of a triangle?

### Formulas for Problem Solving

Some problems are easily solved by applying a formula, such as the Percent Equation or the area of a circle. Formulas are especially common in geometry, and we will see a few of them in this lesson. In many real-world problems, you have to write your own equation, and then solve for the unknown.

#### Let's solve the following problems with formulas:

1. The surface area of a cube is given by the formula \begin{align*}Surface Area=6x^2\end{align*}, where \begin{align*}x=\end{align*} side of the cube. Determine the surface area of a die with a 2-inch side length.

Since \begin{align*}x=2\end{align*} in this problem, we substitute that into the formula for surface area.

\begin{align*}& \text{Surface Area}=6x^2\\ & \text{Surface Area}=6(2)^2\\ & \text{Surface Area}=6\cdot 4=24 \end{align*}

The surface area is 24 square inches.

1. A 500-sheet stack of copy paper is 1.75 inches high. The paper tray on a commercial copy machine holds a two-foot-high stack of paper. Approximately how many sheets is this?

In this situation, we will write an equation using a proportion.

\begin{align*}\frac{\text{number of sheets}}{\text{height}}=\frac{\text{number of sheets}}{\text{height}}\end{align*}

We need to have our heights have the same units, so we will figure out how many inches are in 12 feet. Since 1 foot is 12 inches, then 2 feet are equivalent to 24 inches.

\begin{align*} &\frac{500}{1.75}=\frac{\text{number of sheets}}{24}\\ & \frac{500}{1.75} \times 24=\frac{\text{number of sheets}}{24} \times 24\\ & 6857.14\approx\text{number of sheets}\\ \end{align*}

A two-foot-high stack of paper will be approximately 6857 sheets of paper.

1. The volume of a cone is given by the formula \begin{align*}\text{Volume} = \frac{\pi r^2 (h)}{3}\end{align*}, where \begin{align*}r=\end{align*} the radius, and \begin{align*}h=\end{align*} the height of the cone. Determine the amount of liquid a paper cone can hold with a 1.5-inch diameter and a 5-inch height.

We want to substitute in values for the variables. But first, we are given the diameter, and we need to find the radius. The radius is half of the diameter so \begin{align*}1.5 \div 2=1.5 \times \frac{1}{2}=0.75.\end{align*} \begin{align*}&\text{Volume} = \frac{\pi r^2 (h)}{3}\\ &\text{Volume} = \frac{\pi (0.75)^2 (5)}{3}\\ \end{align*}

We evaluate the expression in our calculator, using \begin{align*}\pi \approx 3.14.\end{align*}

\begin{align*}\text{Volume} \approx 2.94 \end{align*}

The volume of the cone is approximately 2.94 inches cubed. It is approximate because we used an approximation of \begin{align*}\pi.\end{align*}

### Examples

#### Example 1

Earlier, you were asked what the formulas for the area of a circle, the volume of a sphere, and the area of a triangle are.

The formula for the area of a circle is \begin{align*}A = \pi r^2\end{align*} where \begin{align*}r\end{align*} is the radius of the circle and \begin{align*}A\end{align*} is the area.

The formula for the volume of a sphere is \begin{align*}V = \frac{4}{3} \pi r^3\end{align*}  where \begin{align*}r\end{align*} is the radius of the circle and \begin{align*}V\end{align*} is the volume.

The formula for the area of a triangle is \begin{align*}A = \frac{1}{2}bh\end{align*} where \begin{align*}b\end{align*} is the length of the base and \begin{align*}h \end{align*} is the height of the triangle. The height must be perpendicular to the base.

These formulas are important when trying to solve problems in geometry. If you know the formulas, you can substitute in the appropriate values and solve for the necessary variable.

#### Example 2

An architect is designing a room that is going to be twice as long as it is wide. The total square footage of the room is going to be 722 square feet. What are the dimensions in feet of the room?

A formula applies very well to this situation. The formula for the area of a rectangle is \begin{align*}A=l(w)\end{align*}, where \begin{align*}l=\end{align*} length and \begin{align*}w=\end{align*} width. From the situation, we know the length is twice as long as the width. Translating this into an algebraic equation, we get:

\begin{align*}A=(2w)w\end{align*}

Simplifying the equation: \begin{align*}A=2w^2\end{align*}

Substituting the known value for \begin{align*}A\end{align*}: \begin{align*}722=2w^2\end{align*}

\begin{align*}2w^2 & = 722 && \text{Divide both sides by} \ 2. \\ w^2 & = 361 && \text{Take the square root of both sides}. \\ w & = \sqrt{361} = 19 \\ 2w & = 2 \times 19 = 38 \\ w & = 19\end{align*}

The width is 19 feet and the length is 38 feet.

### Review

1. Patricia is building a sandbox for her daughter. It's going to be five feet wide and eight feet long. She wants the height of the sandbox to be four inches above the height of the sand. She has 30 cubic feet of sand. How high should the sandbox be?
2. It was sale day at Macy’s and everything was 20% less than the regular price. Peter bought a pair of shoes, and using a coupon, got an additional 10% off the discounted price. The price he paid for the shoes was $36. How much did the shoes cost originally? 3. Peter is planning to show a video file to the school at graduation, but he's worried that the distance the audience sits from the speakers will cause the sound and the picture to be out of sync. If the audience sits 20 meters from the speakers, what is the delay between the picture and the sound? (The speed of sound in air is 340 meters per second.) 4. Rosa has saved all year and wishes to spend the money she has on new clothes and a vacation. She will spend 30% more on the vacation than on clothes. If she saved$1000 in total, how much money (to the nearest whole dollar) can she spend on the vacation?
5. On a DVD, data is stored between a radius of 2.3 cm and 5.7 cm. Calculate the total area available for data storage in square cm.
6. If a Blu-ray \begin{align*}^{TM}\end{align*} DVD stores 25 gigabytes (GB), what is the storage density in GB per square cm ?
7. The volume of a cone is given by the formula \begin{align*}Volume = \frac{\pi r^2 (h)}{3}\end{align*}, where \begin{align*}r=\end{align*} the radius, and \begin{align*}h=\end{align*} the height of the cone. Determine the amount of liquid a paper cone can hold with a 1.5-inch diameter and a 5-inch height.
8. Consider the conversion \begin{align*}1 \ meter = 39.37 \ inches\end{align*}. How many inches are in a kilometer? (Hint: A kilometer is equal to 1,000 meters.)
9. Yanni’s motorcycle travels \begin{align*}108 \ miles/hour\end{align*}. \begin{align*}1 \ mph = 0.44704 \ meters/second\end{align*}. How many meters did Yanni travel in 45 seconds?
10. The area of a rectangle is given by the formula \begin{align*}A=l(w)\end{align*}. A rectangle has an area of 132 square centimeters and a length of 11 centimeters. What is the perimeter of the rectangle?

Mixed Review

1. Write the following ratio in simplest form: 14:21.
2. Write the following ratio in simplest form: 55:33.
3. Solve for \begin{align*}a:\ \frac{15a}{36} = \frac{45}{12}\end{align*}.
4. Solve for \begin{align*}x:\ \frac{4x+5}{5} = \frac{2x+7}{7}\end{align*}.
5. Solve for \begin{align*}x:\ 4(x-7)+x = 2\end{align*}.
6. What is 24% of 96?
7. Find the sum: \begin{align*}4 \frac{2}{5}- \left (- \frac{7}{3} \right )\end{align*}.

To see the Review answers, open this PDF file and look for section 3.12.

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### Vocabulary Language: English Spanish

TermDefinition
area of a rectangle The formula for the area of a rectangle is given by $A=l(w)$, where $l=$ length and $w=$ width.
surface area of a cube The surface area of a cube is given by the formula $Surface Area=6x^2$, where $x=$ side of the cube.
volume of a cone The volume of a cone is given by the formula $\text{Volume} = \frac{\pi r^2 (h)}{3}$, where $r=$ radius, and $h=$ height of cone.
Formula A formula is a type of equation that shows the relationship between different variables.

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