Problem 5 - College Algebra with Modeling and Visualization

Question

Find the area and perimeter of the rectangle with length \(L\) and width \(W\). \(L=3 x, W=y\)

Step-by-Step Solution

Answer

Area: \(3xy\), Perimeter: \(6x + 2y\).

Step 1: Write the Formula for the Area of a Rectangle

The area of a rectangle is calculated using the formula: \( A = L \times W \) where \(L\) is the length and \(W\) is the width.

Step 2: Substitute Length and Width into Area Formula

Substitute the given expressions for length and width into the area formula: \( A = 3x \times y = 3xy \).

Step 3: Write the Formula for the Perimeter of a Rectangle

The perimeter of a rectangle is calculated using the formula: \( P = 2L + 2W \).

Step 4: Substitute Length and Width into Perimeter Formula

Substitute the given expressions for length and width into the perimeter formula: \( P = 2(3x) + 2(y) = 6x + 2y \).

Step 5: Summary of Results

The area of the rectangle is \( 3xy \) and the perimeter of the rectangle is \( 6x + 2y \).

Key Concepts

Area of a Rectangle Formula

Calculating the area of a rectangle is a fundamental concept in geometry. The area represents the amount of space inside the rectangle, and it is measured in square units. To find this area, you use the formula:

\[ A = L \times W \]
Where:

• \(A\) is the area.
• \(L\) is the length.
• \(W\) is the width.

By multiplying the length by the width, you can determine exactly how much space is enclosed within the rectangle. For example, if the length is expressed as \(3x\) and the width as \(y\), the area is computed as:

\[ A = 3x \times y = 3xy \]
This expression, \(3xy\), is the algebraic representation of the rectangle's area based on the given dimensions.

Perimeter of a Rectangle Formula

The perimeter of a rectangle is the total distance around the rectangle. It is the sum of all its sides, and it is measured in linear units. The formula for finding the perimeter is:

\[ P = 2L + 2W \]
Where:

• \(P\) is the perimeter.
• \(L\) is the length.
• \(W\) is the width.

This formula works by adding together the lengths of all four sides: two lengths and two widths. If the length is \(3x\) and the width is \(y\), substituting these into the formula gives:

\[ P = 2(3x) + 2(y) = 6x + 2y \]
Thus, \(6x + 2y\) represents the perimeter of the rectangle using algebraic expressions for its dimensions.

Algebraic Expressions

Algebraic expressions are a key part of handling formulas involving rectangles, especially when variables represent dimensions. An algebraic expression combines numbers, variables, and operators to describe a mathematical relationship.

In the context of rectangles, variables such as \(x\) and \(y\) are used to represent the dimensions of length and width. For example, with length \(3x\) and width \(y\), the expressions for area and perimeter become:

• Area: \(3xy\)
• Perimeter: \(6x + 2y\)

These expressions allow flexibility and can be used to substitute specific numerical values for \(x\) and \(y\) to find the actual measures of area and perimeter.

Understanding how to manipulate these expressions is crucial for solving various geometric problems and for understanding how changes in dimensions affect the area and perimeter.