Problem 8 - College Algebra with Modeling and Visualization

Question

Find the area and perimeter of the rectangle in terms of the width \(W\). Triple the width \(W\) minus 3 equals the length.

Step-by-Step Solution

Answer

Area: \( 3W^2 - 3W \), Perimeter: \( 8W - 6 \)

Step 1: Understanding the Relationship between Width and Length

From the problem, we know that the length of the rectangle is given by `triple the width minus 3`. We can express this as an equation: \[ L = 3W - 3 \] where \( L \) is the length and \( W \) is the width of the rectangle.

Step 2: Formulate the Perimeter Equation

The perimeter \( P \) of a rectangle is calculated by the formula: \[ P = 2(L + W) \] Substitute the expression for \( L \) from the first step: \[ P = 2((3W - 3) + W) \] which simplifies to: \[ P = 2(4W - 3) \] Therefore, the equation for the perimeter of the rectangle in terms of \( W \) is: \[ P = 8W - 6 \]

Step 3: Formulate the Area Equation

The area \( A \) of a rectangle is given by the formula: \[ A = L \times W \] Substitute the expression for \( L \) from Step 1: \[ A = (3W - 3) \times W \] Simplify to: \[ A = 3W^2 - 3W \] Thus, the area of the rectangle in terms of \( W \) is: \[ A = 3W^2 - 3W \]

Key Concepts

Rectangles

Rectangles are simple yet fascinating geometric shapes. They have opposite sides that are parallel and equal in length. This characteristic shape has four right angles, typically evident by the square corners seen in many daily objects like books or screens.
In algebra, rectangles play a crucial role when working with perimeter and area calculations. Understanding rectangles is not just about knowing the shape, but also exploring the relationships between their sides. Particularly, it's interesting to comprehend how the width and length relate to one another.
For instance, in some problems, relationships might be defined algebraically, such as saying triple the width minus 3 equals the length. This relationship facilitates the creation of mathematical equations, helping us derive various properties of the rectangle.

Perimeter Calculation

Calculating the perimeter of a rectangle is straightforward once you understand its basic properties. The perimeter is the total length around the rectangle. You find it by adding up the lengths of all four sides.
There is a simple formula to calculate the perimeter:

• Identify the width (\( W \)) and the length (\( L \)) of the rectangle.
• Use the formula \( P = 2(L + W) \).

For our specific problem, since the length is defined as three times the width minus 3, you can substitute to find the perimeter in terms of width. So,

• Substitute \( L = 3W - 3 \)
• Simplify \( P = 2((3W - 3) + W) \)
• This results in \( P = 8W - 6 \).

Understanding this allows you to effectively determine how the perimeter changes with different values of width.

Area Calculation

The area of a rectangle represents the total space enclosed within its boundaries. Calculating the area is crucial for understanding how much space is occupied within the shape's borders.
The calculation of a rectangle's area uses the formula:

• Area \( A \) is given by \( A = L \times W \).

In many practical problems, like the one at hand, you are given a relationship between width and length.

• For this problem, substitute \( L = 3W - 3 \).
• You get \( A = (3W - 3) \times W \).
• When expanded, this results in \( A = 3W^2 - 3W \).

This formula helps you comprehend how the area of the rectangle depends on the width alone when the relationship between width and length is algebraically defined. With it, you are able to predict changes in area as the width varies.