Problem 251 - Elementary Algebra

Question

In the following exercises, solve using rectangle properties. The length of a rectangle is 85 feet and the width is 45 feet. What is the perimeter?

Step-by-Step Solution

Answer

The perimeter is 260 feet.

Step 1: Identify Rectangle Properties

A rectangle has opposite sides that are equal in length. The perimeter of a rectangle is the total distance around the outside, which can be found using the formula: \[ P = 2(L + W) \]where \(L\) is the length and \(W\) is the width.

Step 2: Substitute the Given Values

Substitute the given values of length \(L = 85\) feet and width \(W = 45\) feet into the perimeter formula:\[ P = 2(85 + 45) \]

Step 3: Simplify the Expression Inside the Parentheses

First, solve the expression inside the parentheses:\(85 + 45 = 130\)So the equation now is:\[ P = 2 \times 130 \]

Step 4: Calculate the Perimeter

Finally, multiply the result by 2 to find the perimeter:\[ P = 2 \times 130 = 260 \] Therefore, the perimeter of the rectangle is 260 feet.

Key Concepts

perimeter calculation

Understanding how to calculate the perimeter of a rectangle is fundamental in geometry. The perimeter is the total distance around the outside of a rectangle. To find it, you need a specific formula:
\[ P = 2(L + W) \]
Here, P stands for perimeter, L for the length, and W for the width of the rectangle.
Imagine you are walking around the edge of a rectangle. You would walk along both the length and the width, and since a rectangle has two pairs of equal opposite sides, you'd walk each length and each width twice. Hence, the multiplication by 2 in the formula.
To use this formula:

• Step 1: Add the length and the width of the rectangle.
• Step 2: Multiply the result by 2.

For example, if the length of the rectangle is 85 feet and the width is 45 feet, the calculation would look like this:
\[ P = 2(85 + 45) \]
First, calculate inside the parentheses:
\[ 85 + 45 = 130 \]
Then, multiply by 2:
\[ P = 2 \times 130 = 260 \]
So, the perimeter is 260 feet.

length and width relationship

In a rectangle, the opposite sides are equal in length. This is a core property that helps in many calculations related to rectangles. For instance, knowing the length of one side allows you to infer the length of the opposite side. Similarly, knowing the width of one side tells you the width of the opposite side.
Let's denote the length of a rectangle by L and the width by W. This means:

• If L is 85 feet, the opposite side's length is also 85 feet.
• If W is 45 feet, the width on the opposite side is also 45 feet.

This characteristic ensures that you only need the measurements of two adjacent sides to calculate the perimeter and other properties of the rectangle.
Understanding this relationship helps in visualizing problems and organizing the given data efficiently. For example, when using the perimeter formula, you only need the length and width once since the opposite sides are automatically included:
\[ P = 2(L + W) \]
This saves time and reduces potential errors in calculations.

basic algebra

Algebra is a crucial tool in solving many geometric problems. When calculating the perimeter of a rectangle, basic algebraic skills are used to manipulate and solve equations.
To break down our example:
Given the length (L) as 85 feet and the width (W) as 45 feet, the perimeter (P) is found using the formula:
\[ P = 2(L + W) \]
Step 1: Add the length and width:
\[ 85 + 45 = 130 \]
Step 2: Multiply the sum by 2:
\[ 2 \times 130 = 260 \]
So, the perimeter P is 260 feet.
Fundamentally, adding and multiplying are basic algebraic operations. Understanding how to correctly perform these steps ensures accurate results. This problem is a practical application of algebra in geometric contexts, demonstrating how these mathematical disciplines intersect and support one another.