Problem 11 - Elementary Algebra

Question

For Problems 11-32, use the geometric formulas given in this section to help solve the problems. (Objective 3 ) Find the perimeter of a rectangle that is 14 centimeters long and 9 centimeters wide.

Step-by-Step Solution

Answer

The perimeter is 46 cm.

Step 1: Understand the Problem

The problem is asking for the perimeter of a rectangle. Perimeter is the total distance around the rectangle.

Step 2: Recall the Formula for Perimeter of a Rectangle

The perimeter (P) of a rectangle is given by the formula: \( P = 2(l + w) \), where \( l \) is the length and \( w \) is the width.

Step 3: Identify Given Values

The problem states that the length \( l \) is 14 cm and the width \( w \) is 9 cm.

Step 4: Substitute Values into the Formula

Replace \( l \) and \( w \) in the formula with the given numbers: \( P = 2(14 + 9) \).

Step 5: Calculate the Perimeter

First, add the length and width: \( 14 + 9 = 23 \). Then, multiply the sum by 2: \( 2 \times 23 = 46 \).

Step 6: State the Result

The perimeter of the rectangle is 46 centimeters.

Key Concepts

Perimeter of a Rectangle

To find the perimeter of a rectangle, you need to understand what a perimeter is in geometric terms. The perimeter is the total distance around the outside of a shape. It's similar to measuring the borders of a picture frame.
For a rectangle, which has two pairs of equal sides, the formula to calculate the perimeter is given by:

• \( P = 2(l + w) \)

Here, \( l \) stands for the length, and \( w \) stands for the width of the rectangle.
So, to find the perimeter:

• Add the length and the width together.
• Then, multiply the result by 2.

This formula works because a rectangle has opposite sides that are equal, so you're essentially adding up all four sides of the shape.
In our example, substituting the values gave us a perimeter of 46 centimeters. This is how much material you would need to wrap around the rectangle.

Rectangle

A rectangle is a four-sided shape with equal opposite sides and four right angles, making it a very important shape in mathematics and everyday life. Imagine objects like a computer screen, a book, or a piece of paper; they are all typically rectangular.
Here are the key characteristics of a rectangle:

• Two pairs of opposite sides that are equal in length.
• Each of the four angles is a right angle (90 degrees).

Despite its simple appearance, the rectangle is a fundamental shape in geometry, and its properties allow it to be used in many practical and theoretical applications. For instance, it is involved in calculations related to area and perimeter, which are essential concepts in mathematics. Understanding these characteristics helps to grasp more complex geometric ideas later.

Mathematics Education

Mathematics education is about more than just learning formulas; it's about understanding concepts and being able to apply them in various contexts. In this example, understanding the perimeter of a rectangle not only involves knowing the formula \( P = 2(l + w) \) but also understanding why this formula makes sense geometrically.
In learning math, it's crucial to develop:

• Problem-solving skills: For example, identifying what you need to find and the data you have.
• Critical thinking: Breaking down the problem step by step as seen in our solution process.
• Application: Using the formulas in relevant real-world situations.

These skills go beyond the classroom and are important in designing, building, and analyzing various aspects of our world. Education in mathematics empowers students by providing a toolkit for understanding and creating solutions in both theoretical and practical scenarios.