Problem 13 - Algebra and Trigonometry

Question

\(7-18 \cdot\) Express the given quantity in terms of the indicated variable. The area \(\left(\text { in } \mathrm{ft}^{2}\right)\) of a rectangle that is three times as long as it is wide; \(\quad w=\) width of the rectangle (in \(\mathrm{ft} )\)

Step-by-Step Solution

Answer

The area is \(3w^2\).

Step 1: Understanding the Problem

We need to express the area of a rectangle as a function of its width. We know the rectangle is three times as long as it is wide. The variable given is \(w\), which represents the width of the rectangle.

Step 2: Setting up the Relationship

Let the width of the rectangle be \(w\). Since the length is three times the width, the length \(L\) can be expressed as \(L = 3w\).

Step 3: Finding the Area Expression

The area \(A\) of a rectangle is calculated using the formula \(A = \text{length} \times \text{width}\). Substitute the expression for the length: \(A = 3w \times w\).

Step 4: Simplifying the Area Expression

Simplify the expression for the area: \(A = 3w^2\). This represents the area of the rectangle in terms of its width \(w\).

Key Concepts

Rectangle Area

The rectangle area is one of the fundamental concepts in geometry and algebra. It is calculated by multiplying the length and the width of the rectangle. A rectangle is a four-sided shape with opposite sides equal and right angles in each corner. To determine its area, you would simply take the measurement of the length and multiply it by the measurement of the width. This concept is expressed in the formula: \(A = \text{length} \times \text{width}\), where \(A\) represents the area. Understanding how to find the area is crucial, as it helps in various real-world applications, from designing a garden plot to calculating the space within a room.

Variable Representation

Variable representation is an essential algebraic concept that involves using symbols to denote quantities. In this exercise, the variable \(w\) represents the width of the rectangle. This simplifies calculations and allows for the expression of mathematical relationships. Identifying what each variable stands for is key in solving problems effectively.

• Variables act as placeholders and help in generalizing solutions.
• They make complex problems easier to handle by breaking them down into simpler parts.

Using variables like \(w\) streamlines the process of setting up relationships between different elements of a problem, such as length (\(L = 3w\)), making it easier to manipulate and solve for unknowns.

Area Calculation

Area calculation in problems involving variables requires substituting values into an algebraic expression to find the area. Once you've established variable relationships, you need to substitute these into the area formula. For a rectangle whose length is three times its width, you first express the length in terms of the width with \(L = 3w\). Then substitute it into the area equation: \[ A = 3w \times w = 3w^2 \] This formula, \(A = 3w^2\), encapsulates the area in terms of a single variable, \(w\). It's crucial to thoroughly simplify expressions to avoid mistakes and ensure clear, workable solutions. This simplification leads to a deeper understanding of both the problem-solving process and the algebraic techniques needed to reach an accurate solution.