Problem 133 - College Algebra

Question

will help you prepare for the material covered in the next section. If the width of a rectangle is represented by \(x\) and the length is represented by \(x+200,\) write a simplified algebraic expression that models the rectangle's perimeter.

Step-by-Step Solution

Answer

The simplified algebraic expression that models the rectangle's perimeter is \(P = 4x + 400\).

Step 1: Recognize the formula for the perimeter of a rectangle

The formula for a rectangle's perimeter is P = 2L + 2W, where L stands for length and W stands for width of the rectangle.

Step 2: Substitute the given expressions into the formula

Here, the width W is represented by \(x\), and the length L is represented by \(x + 200\). So, the formula becomes P = 2(x + 200) + 2x.

Step 3: Simplify the Algebraic Expression

To simplify the expression, begin by expanding the multiplication and then sum up like terms: P = 2x + 400 + 2x = 4x + 400.

Key Concepts

Perimeter of Rectangle

When calculating the perimeter of a rectangle, think of it as measuring the total distance around the outside of the shape. A rectangle has two lengths and two widths, and the perimeter is the sum of all these sides.

The formula we use is:

• Perimeter (\(P\)) = 2 times the length (\(L\)) plus 2 times the width (\(W\))

So, we write it as \(P = 2L + 2W\).

This formula ensures you include all sides when calculating the total perimeter. Knowing this formula is essential for solving problems related to rectangles in both simple and complex mathematical contexts.

Simplifying Expressions

Simplifying algebraic expressions involves reducing them to their simplest form while preserving their equality. In our exercise, this means taking the long perimeter expression and breaking it down.

First, remember to apply the distributive property effectively. This involves multiplying each term inside the parentheses by the factor outside. For example, when you have \(2(x + 200)\), multiply \(2\) with each term inside. Resulting in \(2x + 400\).

• Step 1: Expand parentheses by distributing.
• Step 2: Combine like terms. Here, add \(2x + 2x\), which gives \(4x\).
• Step 3: Add constant terms separately, leading overall to \(4x + 400\).

By following these steps, you can simplify complex expressions into concise, manageable forms, which makes further calculations easier and less error-prone.

Rectangular Geometry

Rectangular geometry deals with the properties and characteristics of rectangles, which are four-sided shapes with opposite sides that are equal and parallel.

Key features to remember about rectangles include:

• All angles are right angles (90 degrees).
• Opposite sides are equal in length.
• Rectangles are also a type of parallelogram, so they share some properties with them.

Understanding these properties helps in analyzing how a rectangle behaves and relates to its dimensions, particularly when using variables to define side lengths. This knowledge is beneficial when building algebraic expressions and formulas like those used to calculate perimeters or areas.