Problem 10 - College Algebra with Modeling and Visualization

Question

Find the area and perimeter of the rectangle in terms of the width \(W\). The length is 2 less than twice the width \(W\).

Step-by-Step Solution

Answer

Area: \( A = 2W^2 - 2W \); Perimeter: \( P = 6W - 4 \).

Step 1: Define the Length in Terms of Width

The problem states that the length is 2 less than twice the width. We can express the length \( L \) in terms of the width \( W \) using the equation: \( L = 2W - 2 \).

Step 2: Determine the Perimeter Formula

The perimeter \( P \) of a rectangle is calculated by the formula \( P = 2L + 2W \). Substituting for \( L \) from Step 1 gives: \( P = 2(2W - 2) + 2W \).

Step 3: Simplify the Perimeter Equation

Simplify the perimeter equation: \( P = 2(2W - 2) + 2W \). Expanding the equation, we get: \( P = 4W - 4 + 2W \). Combine similar terms: \( P = 6W - 4 \).

Step 4: Determine the Area Formula

The area \( A \) of a rectangle is given by \( A = L \times W \). Using the expression for \( L \) from Step 1, the area equation becomes: \( A = (2W - 2) \times W \).

Step 5: Simplify the Area Equation

Simplify the area equation: \( A = (2W - 2)W \). Distribute the \( W \): \( A = 2W^2 - 2W \). So, the area in terms of \( W \) is \( A = 2W^2 - 2W \).

Key Concepts

Perimeter of a Rectangle

The perimeter of a rectangle is a measure that represents the total length around the rectangle. To find the perimeter, you need to add up all the sides.
In the context of general algebraic expressions, when given variables, it's often more flexible, since you're expressing the perimeter as a formula rather than a set number.
To calculate the perimeter of a rectangle, we use the formula:

• \( P = 2L + 2W \)

Where \( L \) is the length and \( W \) is the width.
This formula shows how the length and width are each counted twice since a rectangle has two pairs of equal sides.
In our specific example, the length is defined in terms of \( W \) (width): \( L = 2W - 2 \).
By plugging \( L \) into the perimeter formula, we solve it step-by-step:

• Expand: \( P = 2(2W - 2) + 2W \)
• Simplify: \( P = 4W - 4 + 2W \)
• Combine like terms: \( P = 6W - 4 \)

The simplicity of using variables makes it possible to use this formula for any width to find a perimeter.

Area of a Rectangle

Finding the area of a rectangle is about measuring the amount of space contained within its boundaries. The area helps understand how much surface the rectangle covers.
For any rectangle, the area is calculated using the formula:

• \( A = L \times W \)

Where \( A \) stands for the area, \( L \) for length, and \( W \) for width.
This method, which relies on multiplication, easily accommodates situations with algebraic expressions or variables.
In our exercise, we replaced \( L \) with \( 2W - 2 \) to express it in terms of width \( W \):

• Substitute and simplify: \( A = (2W - 2) \times W \)
• Distribute \( W \): \( A = 2W^2 - 2W \)

This formula gives you the area of the rectangle based on its width, allowing you to adjust the width and recalibrate the area quickly. It demonstrates a fine example of using algebraic expressions for real-world applications.

Variables and Expressions

Variables and expressions are fundamental concepts in algebra that allow us to represent mathematical ideas flexibly.
When we talk about algebraic expressions, we refer to a combination of numbers, variables, and arithmetic operations.
Variables, like \( W \) in our exercise, stand in for unknown values which can vary. An expression like \( 2W - 2 \) shows a variable in action, determining the length of a rectangle by operating on the width.
This allows us to perform calculations without a specific numeric value initially. Here’s how we see this in action:

• The expression \( 2W - 2 \) adjusts based on different widths.
• This approach tackles problems where direct measurement might be impossible or impractical early on.

By using both variables and expressions, you can create formulas to solve problems abstractly then apply them concretely by substituting numerical values for variables.
This versatility is what makes algebra an essential tool in everyday math.