Problem 65 - Prealgebra

Question

Find the area and perimeter for a rectangle if the length and width are as given below. \(I=25\) feet, \(w=12\) feet

Step-by-Step Solution

Answer

The area is 300 square feet and the perimeter is 74 feet.

Step 1: Understand Rectangle Properties

The area of a rectangle is calculated using the formula: \( A = l \times w \), where \( l \) is the length and \( w \) is the width. The perimeter of the rectangle is calculated using: \( P = 2 \, (l + w) \).

Step 2: Substitute Values into Area Formula

Substitute the given length \( l = 25 \) feet and width \( w = 12 \) feet into the area formula: \( A = 25 \times 12 \).

Step 3: Calculate the Area

Compute the product for the area: \( A = 25 \times 12 = 300 \) square feet. The area of the rectangle is \( 300 \) square feet.

Step 4: Substitute Values into Perimeter Formula

Use the perimeter formula with \( l = 25 \) feet and \( w = 12 \) feet: \( P = 2 \, (25 + 12) \).

Step 5: Calculate the Perimeter

First calculate the sum inside the parentheses: \( 25 + 12 = 37 \) feet. Then compute the perimeter: \( P = 2 \, \times 37 = 74 \) feet. Hence, the perimeter of the rectangle is \( 74 \) feet.

Key Concepts

Area of a Rectangle

Rectangles are four-sided shapes with opposite sides that are equal in length. To find the area of a rectangle, you multiply its length by its width. The formula looks like this: \( A = l \times w \). Here, \( A \) stands for area, \( l \) represents the length, and \( w \) is the width.

This calculation tells us how much space is inside the rectangle. Imagine painting a wall or covering a field — the area gives you that coverage size.

In our exercise, we have a rectangle with a length of 25 feet and a width of 12 feet. By using the area formula:

• Plug in the length: \( l = 25 \) feet.
• Plug in the width: \( w = 12 \) feet.
• Calculate: \( A = 25 \times 12 = 300 \; \text{square feet} \).

So, the rectangle's area is 300 square feet. This means if we wanted to cover this rectangle entirely, we would need 300 square feet of materials.

Perimeter of a Rectangle

The perimeter of a rectangle is the total distance around the outside of the shape. If you walk around a field, tracing its edges, you are essentially measuring its perimeter. To find it, you add up all the sides or use the simplified formula: \( P = 2 \times (l + w) \).

This formula works because a rectangle's opposite sides are equal. Instead of adding all four sides separately, you can simply double the sum of the length and width.

For our rectangle with a length of 25 feet and width of 12 feet:

• Add the length and width: \( 25 + 12 = 37 \) feet.
• Double that result: \( P = 2 \times 37 = 74 \; \text{feet} \).

Therefore, the perimeter is 74 feet. This number tells us how much fencing we would need to enclose the rectangle completely.

Elementary Math

Elementary math serves as the foundation for understanding mathematics as a whole. It introduces basic concepts like addition, subtraction, multiplication, and division, which are essential for solving problems related to shapes and measurements.

Understanding the formulas for area and perimeter is a great example of applying basic math. The ability to multiply and add accurately allows you to determine the size and boundary of different geometrical figures such as rectangles, which is important in everyday tasks like construction or crafting.

Remember these crucial concepts of elementary math when working with rectangles:

• Area: This involves multiplication and gives the space inside a rectangle.
• Perimeter: This entails both addition and multiplication and tells how long the outline of the shape is.

Mastering these basic operations helps you approach more complex problems with confidence.