Problem 75 - Elementary Algebra

Question

The length of a rectangle is \(8 \mathrm{in}\). longer than the width. a. If \(W=\) width, write a polynomial expression in \(W\) that represents the length, and draw a diagram of the rectangle. Do not include the units. b. Write a polynomial expression in \(W\) that represents the perimeter. c. Write a polynomial expression in \(W\) that represents the area.

Step-by-Step Solution

Answer

The length is \(W + 8\). The perimeter is \(4W + 16\). The area is \(W^2 + 8W\).

Step 1 - Express the Length in Terms of Width

Given that the length of the rectangle is 8 inches longer than the width, let the width be denoted by the variable \(W\). Thus, the length can be written as \(W + 8\).

Step 2 - Draw the Diagram of the Rectangle

Draw a rectangle and label the width as \(W\) and the length as \(W + 8\). This visual representation will help in understanding the relationship better.

Step 3 - Write the Polynomial for the Perimeter

The formula for the perimeter of a rectangle is given by:\[ P = 2 \times (\text{Length} + \text{Width}) \]Substitute \(W + 8\) for the length and \(W\) for the width:\[ P = 2 \times ((W + 8) + W) \]Simplify the expression:\[ P = 2 \times (2W + 8) \]\[ P = 4W + 16 \]

Step 4 - Write the Polynomial for the Area

The formula for the area of a rectangle is given by:\[ A = \text{Length} \times \text{Width} \]Substitute \(W + 8\) for the length and \(W\) for the width:\[ A = (W + 8) \times W \]Simplify the expression:\[ A = W^2 + 8W \]

Key Concepts

Polynomial Expressions

A polynomial is an expression consisting of variables, coefficients, and arithmetic operations such as addition, subtraction, and multiplication. In elementary algebra, polynomial expressions commonly appear in problems involving shapes and equations. For example, if we are calculating lengths or areas in terms of a variable, we often use polynomials.

In our exercise, we used the variable \(W\) to denote the width of a rectangle. The length of the rectangle was given as 8 inches longer than the width, which can be written as the polynomial expression \(W + 8\). This type of polynomial expression is simple as it consists only of terms involving \(W\) and constants.

When dealing with polynomials, we often simplify them by combining like terms and performing arithmetic operations. For example, in the case of the perimeter (detailed below), multiple steps involve simplifying polynomial expressions to find our final answer.

Rectangle Perimeter

The perimeter of a rectangle is the total distance around the outside of the rectangle. The formula to calculate the perimeter \(P\) of a rectangle is given by:
\[ P = 2 \times (\text{Length} + \text{Width}) \]

In our exercise, since we know the length is expressed as \(W + 8\) and the width is \(W\), substituting these expressions into the perimeter formula gives us:
\[ P = 2 \times ((W + 8) + W) = 2 \times (2W + 8) = 4W + 16 \]
This polynomial \(4W + 16\) represents the perimeter of the rectangle as a function of its width. Simplifying these expressions step-by-step makes it clearer to see the contributions from each side of the rectangle into the overall perimeter.

Understanding how to manipulate and simplify such polynomial expressions is crucial in solving algebraic problems related to geometric figures.

Rectangle Area

The area of a rectangle is given by the amount of space enclosed within its sides. The formula for the area \(A\) of a rectangle is:
\[ A = \text{Length} \times \text{Width} \]

Again, in our exercise, the length is \(W + 8\) and the width is \(W\). Substituting these values into the area formula, we get:
\[ A = (W + 8) \times W \]
Expanding the expression gives us:
\[ A = W^2 + 8W \]
This polynomial \(W^2 + 8W\) represents the area of the rectangle in terms of its width. Polynomials like this one often appear in problems requiring the calculation of areas or volumes because they make it easier to understand how different dimensions of a shape contribute to its total size.

By practicing these polynomial operations, students can become more comfortable with algebraic manipulations, which are essential in solving more complex problems later on.