Problem 87 - Elementary Algebra

Question

A drafter is making enlargements of a rectangular drawing that preserve the relative width and length of the drawing. The length of the drawing is five- fourths of 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

Length: \(\frac{5}{4}W\), Perimeter: \(\frac{9}{2}W\), Area: \(\frac{5}{4}W^2\).

Step 1: Express the Length in terms of Width

Given that the length is five-fourths of the width, the polynomial expression for the length in terms of the width, denoted as W, is \[L = \frac{5}{4}W\].

Step 2: Draw a Diagram

Draw a rectangle and label the width as W and the length as \(\frac{5}{4}W\). This will help visualize the problem.

Step 3: Find the Perimeter

The perimeter of a rectangle is given by the formula \[P = 2L + 2W\]. Substituting the expression for length, we get \[P = 2 \left(\frac{5}{4}W\right) + 2W = \frac{10}{4}W + 2W = \frac{10}{4}W + \frac{8}{4}W = \frac{18}{4}W = \frac{9}{2}W\].

Step 4: Find the Area

The area of a rectangle is given by the formula \(A = L \times W\). Substituting the expression for length, we get \[A = \left( \frac{5}{4}W \right) \times W = \frac{5}{4}W^2\].

Key Concepts

Polynomial Expressions

Polynomial expressions are algebraic expressions that consist of variables and coefficients. They involve operations like addition, subtraction, multiplication, and non-negative integer exponents of variables.
In this problem, we use polynomial expressions to represent the length, perimeter, and area of a rectangle. Such expressions help us perform geometric calculations algebraically.
For example, we express the length of the rectangle as \(L = \frac{5}{4}W\), where \(W\) is the width. These expressions allow us to further calculate important properties like perimeter and area using algebraic methods.

Rectangle Geometry

A rectangle is a four-sided polygon with opposite sides equal in length and every angle a right angle (90 degrees). The basic properties of a rectangle include:

• Two pairs of opposite, equal, and parallel sides.
• Four right angles.
• Diagonals that bisect each other.

For geometric problems involving rectangles, understanding these properties is crucial.
In this exercise, the length of the drawing is five-fourths of the width which can be visually represented in a diagram. Just sketch a rectangle, label one side as \(W\) and the opposite longer side as \( \frac{5}{4}W\).

Rectangular Perimeter and Area Calculations

The perimeter of a rectangle is the total distance around the outside, calculated by adding the lengths of all four sides together. The formula is:
\[ P = 2L + 2W \]
Substituting \(L = \frac{5}{4}W\), the perimeter becomes:
\[ P = 2 \times \frac{5}{4}W + 2W = \frac{10}{4}W + \frac{8}{4}W = \frac{18}{4}W = \frac{9}{2}W \]
The area represents the amount of space inside the rectangle, calculated by multiplying the length and the width. The formula is:
\[ A = L \times W \]
With \(L = \frac{5}{4}W\), the area translates to:
\[ A = \frac{5}{4}W \times W = \frac{5}{4}W^2 \]
Understanding these formulas and how to use polynomial expressions in them ensures accurate rectangular perimeter and area calculations, making it easier to solve geometry problems efficiently.