Problem 253 - Elementary Algebra

Question

In the following exercises, solve using rectangle properties. A rectangular room is 15 feet wide by 14 feet long. What is its perimeter?

Step-by-Step Solution

Answer

The perimeter is 58 feet.

Step 1 - Identify the Given Measurements

We are given that the width of the rectangular room is 15 feet and the length is 14 feet.

Step 2 - Recall the Perimeter Formula for a Rectangle

The formula for the perimeter of a rectangle is given by \[ P = 2 \times ( \text{length} + \text{width} ) \]

Step 3 - Substitute the Given Values into the Formula

Substitute the length and width into the formula: \[ P = 2 \times ( 14 \text{ feet} + 15 \text{ feet} ) \]

Step 4 - Perform the Calculations

First, add the length and width: \[ 14 \text{ feet} + 15 \text{ feet} = 29 \text{ feet} \] Then multiply by 2: \[ P = 2 \times 29 \text{ feet} = 58 \text{ feet} \]

Step 5 - State the Final Perimeter

The perimeter of the rectangular room is 58 feet.

Key Concepts

Perimeter Calculation

When we talk about the perimeter of a shape, we mean the total distance around the shape. For a rectangle, the perimeter is calculated by adding up all four sides. However, because opposite sides of a rectangle are equal, a quick way to find the perimeter is to use the formula: \[ P = 2 \times ( \text{length} + \text{width} ) \]This formula allows us to sum the length and width once, then multiply the result by 2 to account for all four sides.

• First, sum the lengths of one side and one width.
• Next, multiply that sum by 2 to get the total perimeter.

For example, in our exercise, we have a length of 14 feet and a width of 15 feet. We substitute these values into our formula like this: \[ P = 2 \times ( 14 \text{ feet} + 15 \text{ feet} ) \] Performing the calculations, we first add 14 feet and 15 feet to get 29 feet. Then, we multiply by 2 to get the final perimeter of 58 feet.

Rectangle Properties

A rectangle is a simple yet important geometric shape with some unique properties. Here are a few essential properties to understand:

• Opposite sides are equal in length.
• All angles are right angles (90 degrees).
• The length is often labeled as the longer side, while the width is the shorter side.

Because opposite sides are equal, it simplifies many calculations. For example, if you know the length and width, you automatically know the measurements of all four sides. This makes it easier to calculate things like perimeter and area. In our exercise, knowing the room’s length and width (14 feet and 15 feet, respectively) allows us to easily apply the perimeter formula.

Basic Algebra

Basic algebra is all about using symbols and numbers to solve problems. When dealing with geometric shapes like rectangles, algebra helps us organize our calculations and find unknown values. The general steps involved are:

• Identify the given values.
• Recall or find the relevant formula.
• Substitute the given values into the formula.
• Perform the necessary operations to solve the problem.

In the case of our rectangular room, basic algebra steps might look like this:

1. Given values: width (15 feet), length (14 feet).
2. Recall the perimeter formula for a rectangle: \[ P = 2 \times ( \text{length} + \text{width} ) \].
3. Substitute the values: \[ P = 2 \times ( 14 \text{ feet} + 15 \text{ feet} ) \].
4. Calculate the sum inside the parentheses: \[ 14 + 15 = 29 \] feet.
5. Multiply by 2: \[ 2 \times 29 \text{ feet} = 58 \text{ feet} \]
6. Result: The perimeter is 58 feet.

Understanding these basic algebra steps makes it easier to handle a variety of mathematical problems, not just geometry-related ones.