Question

Photography The outer edges of a picture frame are 21 inches by 15 inches. The sides of the frame are 2 inches wide. (Lesson 1-6) a. Draw a picture to represent the frame. Label all the information presented in the problem. b. Find the area of a picture that will show in this frame.

Short answer

The area of the picture that shows is 187 square inches.

Step-by-step solution

Understand the Problem

We are given the dimensions of a picture frame: 21 inches by 15 inches. The frame itself is 2 inches wide on all sides. We need to find the dimensions and the area of just the picture (excluding the frame).

Determine Picture Dimensions

The frame is 2 inches wide, meaning it encroaches 2 inches from each side. Therefore, the inner dimensions of the picture can be obtained by subtracting 4 inches from both the width and the height (as the frame is on both sides). So, the picture's dimensions are: \( 21 - 4 = 17 \) inches by \( 15 - 4 = 11 \) inches.

Calculate Picture Area

The area of the picture (the part visible within the frame) is calculated by multiplying the adjusted dimensions: \( 17 \) inches by \( 11 \) inches. Thus, the area that shows inside the frame is \( 17 \times 11 = 187 \) square inches.

Key concepts

Measurements

Measurements are the foundation of solving any geometry problem. They help us establish the framework and dimensions we need to work with. Let's go through the measuring concept as it applies to a picture frame.
First, you need to visualize the situation: Imagine the picture frame as two rectangles, one larger and one smaller. The larger rectangle includes the frame's width, while the smaller one is just the picture inside. The problem specifies that the entire frame is 21 inches by 15 inches. Additionally, the sides of the frame itself are 2 inches wide.
By subtracting twice the frame width (since it affects both sides of length and width), we derive the actual picture's dimensions. This is a good reminder:

• Subtracting twice the frame size from the length: 21 inches - 4 inches = 17 inches
• Subtracting twice the frame size from the width: 15 inches - 4 inches = 11 inches

Measuring correctly sets up the rest of the solution. So, always take care with these numbers at the start!

Area Calculation

Calculating the area is a primary skill needed in geometry. The goal is to find the space within the boundaries you have measured. In this case, that means the area of just the picture within the frame. After subtracting the frame, our measurements give the dimensions of the picture as 17 inches by 11 inches.
The formula for the area of a rectangle is simple: multiply the length by the width.
Let's break that down for clarity:

• Picture Dimensions: Length = 17 inches and Width = 11 inches
• Area Calculation: \\[ 17 \times 11 = 187 \\] The result is an area of 187 square inches.

Area calculations provide us a way to understand the size of a surface. In practical terms, it tells us how much of the picture will be visible. This type of calculation is handy in various real-life applications, from determining paint quantities to assessing usable space.

Mathematical Reasoning

Mathematical reasoning is all about understanding why and how to apply certain methods to solve problems. Let's use this skill with our picture frame exercise.
Here, you reason out that the extra frame width affects the dimensions of the picture itself. This understanding allows you to subtract accurately and find your needed numbers.
Adjusting measurements to include or exclude frames becomes clear with reasoning, following steps logically:

• Recognize that a frame encroaches equally from both sides of a dimension.
• Apply subtraction to both dimensions once for each side affected.
• Always review the logical flow of reasoning steps to ensure each part follows the previous one correctly.

Reasoning teaches you how to approach problems systematically, ensuring no detail or step is overlooked. It strengthens your problem-solving skills, enabling you to handle complex geometry tasks with confidence. This is particularly useful in spaces where accuracy is crucial, such as engineering or design.