Problem 90 - Elementary Algebra

Question

A rectangular parking lot is three times as long as it is wide. 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

a. Length (L) = 3W; b. Perimeter (P) = 8W; c. Area (A) = 3W^2.

Step 1: Define the given relationships

The problem states that the length of the parking lot is three times the width. Let's define the width as W. Therefore, the length can be written as 3W.

Step 2: Write the polynomial expression for the length

Using the relationship given, we can express the length (L) as follows: L = 3W

Step 3: Draw a diagram of the rectangle

To visualize the parking lot, draw a rectangle and label the width as W and the length as 3W. This will help in understanding the geometric relationships. (Diagram is not included in text form.)

Step 4: Write the polynomial expression for the perimeter

The perimeter (P) of a rectangle is given by the formula P = 2L + 2W. Substitute the expression for the length from step 2: \[P = 2(3W) + 2(W) = 6W + 2W = 8W\]

Step 5: Write the polynomial expression for the area

The area (A) of a rectangle is given by the formula A = L * W. Substitute the expression for the length from step 2: \[A = (3W) * W = 3W^2\]

Key Concepts

rectangular dimensions

When working with problems related to rectangular shapes, it is important to understand the basic dimensions involved. A rectangle has two essential measurements: the length and the width. In our exercise, the parking lot's length is defined as three times its width. By defining the width as a variable, say, \(W\), we can easily express the length as \(3W\). Visualizing these relationships can often help.
Writing out the variables and their relationships provides a clear starting point for solving more complex questions like perimeter and area. Physically drawing the dimensions in a diagram also helps solidify your understanding of the problem structure.

polynomial expressions

Polynomial expressions are a key part of solving algebraic equations involving geometric shapes. In our example, we used a polynomial to represent both the length and width of the rectangle. A polynomial is a mathematical expression involving a sum of powers in one or more variables, multiplied by coefficients. Here, the length was represented as \(3W\), which is already a simple polynomial with just one term. Simplifying polynomial expressions correctly helps in forming accurate equations for other properties, like perimeter and area, which are essential for solving geometry-related algebra problems.
Understanding how to form and simplify polynomial expressions makes it easier to handle real-world scenarios related to measurements.

perimeter and area formulas

Formulas for perimeter and area are fundamental in geometry. The perimeter of a rectangle is obtained by adding twice its length and twice its width. For our parking lot, this becomes:
\[ P = 2(3W) + 2(W) = 6W + 2W = 8W \]
The area of a rectangle is found by multiplying its length by its width:
\[ A = (3W) * W = 3W^2 \]
These expressions give us the total perimeter and area in terms of the width. Inserting dimensions into these formulas provides the values required for practical applications, such as fencing a parking lot (perimeter) or paving it (area). Mastering these formulas ensures that you can accurately compute necessary values for any rectangular shape.

basic algebra

Algebra serves as a foundation for solving geometry problems. Key concepts include understanding variables, ratios, polynomial expressions, and basic arithmetic operations. For example, defining the width as \(W\) and recognizing that the length is three times the width translates into the algebraic format \(3W\).
Simplifying these polynomial expressions for the perimeter and area leverages fundamental algebraic techniques. This includes combining like terms and performing multiplication. Basic algebra allows you to turn geometric statements into solvable equations.
Grasping these algebraic principles forms the bedrock of more advanced mathematics and helps you solve a wide array of real-world mathematical problems.