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Chapter 2: Problem 62

Calculate the perimeter of an 8 -by-10-inch picture.

Short Answer

Expert verified
The perimeter of the picture is 36 inches.

Step by step solution

01

Understand the Perimeter Formula

The perimeter of a rectangle is calculated using the formula \( P = 2 \times (l + w) \), where \( l \) is the length and \( w \) is the width of the rectangle.
02

Identify the Rectangle's Measurements

In this problem, the picture is an 8-by-10-inch rectangle. Here, \( l = 10 \text{ inches} \) and \( w = 8 \text{ inches} \).
03

Substitute the Measurements into the Formula

Substitute the values into the perimeter formula: \( P = 2 \times (10 + 8) \).
04

Perform the Addition Inside the Parentheses

Calculate the sum inside the parentheses: \( 10 + 8 = 18 \).
05

Multiply by 2

Now, multiply the result by 2: \( P = 2 \times 18 = 36 \).
06

State the Perimeter

The perimeter of the picture is 36 inches.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Rectangle Measurements
Understanding rectangle measurements is crucial for calculating the perimeter. A rectangle is a four-sided shape, characterized by having opposite sides that are equal in length and forming right angles at each corner.
In the context of a rectangle:
  • Length (\( l \)) is usually referred to as the longer side.
  • Width (\( w \)) is the shorter side or the one perpendicular to the length.
Measurements are typically given in consistent units, such as inches or centimeters, to ensure accuracy in calculations. In our example, the picture has the dimensions of 8 inches by 10 inches. The length is 10 inches, and the width is 8 inches. Knowing these measurements allows us to properly prepare for the next steps in finding the perimeter.
Perimeter Formula
The perimeter of a rectangle is the total length around the rectangle, which means adding up all four sides. The formula to calculate the perimeter (\( P \)) is given by:\[P = 2 \times (l + w)\]where "2" accounts for adding both pairs of the rectangle's sides. This formula helps in deriving the perimeter no matter the rectangle's size, as long as the length (\( l \)) and width (\( w \)) are known. By replacing the variables in the formula with the rectangle's actual measurements, you can efficiently find the perimeter. In our problem, using 10 inches for length and 8 inches for width demonstrates how direct application of this formula solves for the perimeter.
Step by Step Calculation
Let's go through the step-by-step calculation to find the perimeter, ensuring comprehension of each action:
  • Step 1: Identify Values - Establish the rectangle's measurements: length (\( l = 10 \) inches) and width (\( w = 8 \) inches).
  • Step 2: Apply the Formula - Substitute these values into the formula: \( P = 2 \times (10 + 8) \).
  • Step 3: Simplify Inside the Parentheses - First, carry out the operation inside the parentheses: \( 10 + 8 = 18 \).
  • Step 4: Final Multiplication - Multiply the result by 2: \( 2 \times 18 = 36 \).
  • Conclusion - The calculated perimeter of the rectangle, or the picture, is 36 inches.
This systematic approach ensures complete understanding, making it easier to tackle similar problems in the future.

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