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3.8: Problem-Solving Strategies: Use a Formula

Difficulty Level: At Grade Created by: CK-12
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Some problems are easily solved by applying a formula, such as the Percent Equation or the area of a circle. In this lesson, you will include using formulas in your toolbox of problem-solving strategies.

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?

This situation applies very well to a formula. 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:


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.

Practice Set

Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set.  However, the practice exercise is the same in both. CK-12 Basic Algebra: Word Problem Solving 3 (11:06)

  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. 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?
  3. 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?
  4. 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).
  5. 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?
  6. 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.
  7. If a Blu-ray \begin{align*}^{TM}\end{align*} DVD stores 25 gigabytes (GB), what is the storage density, in GB per square cm ?
  8. 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*} radius, and \begin{align*}h=\end{align*} height of cone. Determine the amount of liquid a paper cone can hold with a 1.5-inch diameter and a 5-inch height.
  9. 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)
  10. 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?
  11. 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?
  12. 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 1-inch side length.

Mixed Review

  1. Write the ratio in simplest form: 14:21.
  2. Write the 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*}y:\ 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*}.

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