Problem 23 - Intermediate Algebra

Question

Use an area formula to find the unknown measurement. See Example 2 Wrapping Presents. There are \(30 \mathrm{ft}^{2}\) of gift wrapping paper on a roll. When unrolled, the sheet of paper is 12 feet long. What is its width?

Step-by-Step Solution

Answer

The width of the wrapping paper is 2.5 feet.

Step 1: Understand the Area Formula

To find the area of a rectangle, we use the formula \( A = \text{length} \times \text{width} \). For this problem, you are given that the area \( A = 30 \text{ ft}^2 \) and the length is 12 feet. We need to find the width.

Step 2: Rearrange the Formula to Find Width

Since we know the area and the length, we can rearrange the area formula to solve for the width:\[ \text{width} = \frac{A}{\text{length}} \]

Step 3: Substitute the Known Values

Substitute the known values into the rearranged formula: \[ \text{width} = \frac{30 \text{ ft}^2}{12 \text{ ft}} \]

Step 4: Calculate the Width

Perform the division to calculate the width: \[ \text{width} = \frac{30}{12} = 2.5 \text{ ft} \]

Step 5: Conclusion

The width of the wrapping paper is 2.5 feet.

Key Concepts

Area Formula

The area of a rectangle is calculated by multiplying the length by the width. This fundamental formula is expressed as \( A = \text{length} \times \text{width} \).

Knowing this formula is crucial, especially if you're dealing with rectangular shapes like wrapping paper or even flooring tiles. In our exercise with the wrapping paper, the area given is 30 square feet, with a specified length of 12 feet. Therefore, the task becomes finding out the unknown dimension - the width.

By rearranging the area formula, you can solve for either the length or width, provided you have the other dimension and the area. This rearrangement and manipulation of the formula is a common aspect of algebra.

Algebraic Expressions

Algebraic expressions are powerful tools that allow us to describe and manipulate relationships between numbers. In the rectangle problem, the area formula \( A = \text{length} \times \text{width} \) serves as an algebraic expression where the goal is to find the unknown width.

By restructuring the formula, we extract the specific formula needed to find the unknown: \( \text{width} = \frac{A}{\text{length}} \). This operation is a simple example of solving an algebraic equation, which frequently arises in real-world problems.

Understanding how to manipulate these expressions provides the capability to solve for unknown variables efficiently and is a foundational skill in problem solving throughout mathematics.

Problem-Solving Steps

Effective problem-solving often involves systematic steps to ensure clarity and precision. Using our rectangle's area question, the steps were divided simply and clearly:

• First, grasp the problem and the formula needed.
• Second, rearrange the formula to isolate what you need to find. This step is critical and involves algebraic manipulations.
• Third, plug in the known values and calculate. Substitution into the equation places the knowns and simplification follows.
• Finally, interpret the solution. This careful conclusion ensures that the calculated width is sensibly applicable within the context of the problem.

This structured approach to tackling mathematical problems not only facilitates understanding but also builds the confidence needed to solve more complex issues in mathematics or even in varied real-life scenarios.