Problem 7 - College Algebra with Modeling and Visualization

Question

Find the area and perimeter of the rectangle in terms of the width \(W\). The width \(W\) is half the length.

Step-by-Step Solution

Answer

Area = \(2W^2\); Perimeter = \(6W\).

Step 1: Define the length in terms of width

According to the problem, the width \( W \) is half the length. Thus, the length \( L \) can be expressed as \( L = 2W \).

Step 2: Calculate the area of the rectangle

The area of a rectangle is given by the formula \( \text{Area} = \text{Length} \times \text{Width} \). Substituting the expression for the length, the area becomes \( \text{Area} = 2W \times W = 2W^2 \).

Step 3: Calculate the perimeter of the rectangle

The perimeter of a rectangle is given by the formula \( \text{Perimeter} = 2L + 2W \). Substituting the expression for the length, the perimeter becomes \( \text{Perimeter} = 2(2W) + 2W = 4W + 2W = 6W \).

Step 4: Conclusion

The area and perimeter of the rectangle in terms of the width \( W \) are \( 2W^2 \) and \( 6W \) respectively.

Key Concepts

Understanding the Area of a Rectangle

The area of a rectangle is the amount of space the rectangle covers on a flat surface. Imagine coloring the entire rectangle with your favorite color. That's the area! To find this, you multiply the length by the width. When we're told that width \( W \) is half of the length, we look for a way to express the length using \( W \) as well. Here, the length \( L \) is double the width. Mathematically, that's written as \( L = 2W \). Using the area formula \( \text{Area} = \text{Length} \times \text{Width} \):

• Substitute the length with \( 2W \).
• Compute the area: \( \text{Area} = 2W \times W = 2W^2 \).

This result tells you the area in terms of the width. It's a good start to understanding how changes in width affect the space inside the rectangle.

Calculating the Perimeter of a Rectangle

The perimeter is like the belt around the waist of the rectangle. It's the total distance you travel if you walk around the edges once. To find the perimeter, we add all the sides' lengths. The formula is \( \text{Perimeter} = 2L + 2W \). Again, with \( L = 2W \), we can substitute into the perimeter formula:

• Replace \( L \) with \( 2W \): \( \text{Perimeter} = 2(2W) + 2W \).
• Compute step by step: \( \text{Perimeter} = 4W + 2W = 6W \).

Now, you know how far you would walk to go around the rectangle. Understanding this helps visualize how perimeter increases with wider or longer rectangles.

Using Algebraic Expressions with Rectangles

Algebraic expressions are like recipes for calculations: they show how to compute things step by step using symbols and numbers. In this exercise, you express both the area and the perimeter of a rectangle using the variable \( W \), which represents the width.Understanding algebraic expressions:

• They allow you to generalize rectangle properties using variables.
• They make calculations easier and flexible for different values of \( W \).

In this case:- You express length as an algebraic expression of width: \( L = 2W \).- You use these expressions in standard formulas. For area, you get \( 2W^2 \), and for perimeter, you obtain \( 6W \).Algebraic expressions simplify complex mathematical ideas, helping us see patterns and understand relationships between different quantities.