Problem 13 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

The area of the rectangle is \(3w^2\) square feet.

Step 1: Understanding the Problem

We are given a rectangle whose length is three times its width. We need to express the area of this rectangle using the width, denoted as \( w \). The formula for the area of a rectangle is given by the product of its length and width.

Step 2: Expressing Variables

The problem states that the length \( L \) of the rectangle is three times the width \( w \). Therefore, we can express the length as \( L = 3w \).

Step 3: Substituting Variables into the Area Formula

The standard formula for the area \( A \) of a rectangle is \( A = ext{length} imes ext{width} \). Substituting the values we have: \( A = L imes w = 3w imes w \).

Step 4: Solving the Expression

After substituting, we have the expression for area as \( A = 3w imes w = 3w^2 \). Thus, the area of the rectangle, in terms of its width, is \( 3w^2 \).

Key Concepts

Width of Rectangle

The width of a rectangle is one of its key dimensions, along with its length. When we talk about the width, it refers to the shorter side of the rectangle. In a mathematical problem, like the one we're dealing with, the width is often represented by a variable. Here, the width is denoted as \( w \) and is typically measured in standard units such as feet or meters.
Understanding the width is crucial because it is directly involved in calculating the area of the rectangle. The area provides a measure of how much surface the rectangle covers. By knowing the width and using a simple formula, we can determine the space within the rectangle's boundaries.
Keep in mind that the width doesn't change unless specified, making it a constant variable in most calculations involving rectangles.

Length and Width Relationship

In this problem, we're given an important relationship between the rectangle’s length and its width. It's stated that the length is three times the width. This relationship helps us express the length in terms of the width.
To do this, we use the formula: \( L = 3w \). This equation tells us that for every unit of width \( w \), the length \( L \) is thrice as much. So, if you know the width, you can directly calculate the length by simply tripling the width.
Such relationships are common in geometry problems, allowing us to express one dimension in terms of another, simplifying calculations and making complex relationships easier to visualize.

Expressing Area in Terms of a Variable

Once we establish the relationship between length and width, we can express the area of the rectangle using these variables. The area \( A \) of a rectangle is found by multiplying its length by its width, typically expressed as \( A = \text{length} \times \text{width} \).
In our problem, knowing the length is \( L = 3w \), we can substitute this into the area formula, resulting in:

• \( A = L \times w \)
• \( A = 3w \times w \)
• \( A = 3w^2 \)

By multiplying \( 3w \) with \( w \), we derive that the area in terms of its width \( w \) is \( 3w^2 \).
This expression shows how powerful expressing dimensions in terms of a variable can be. It allows simplification and precision when conducting mathematical operations, ensuring all parts are correctly accounted for in terms of the initial parameters.