A 5-inch By 7-inch Photograph Is Placed Inside A Picture Frame. Both The Length And Width Of The Frame Are $2x$ Inches Larger Than The Width And Length Of The Photograph. Which Expression Represents The Perimeter Of The Frame?A. $4x +

Understanding the Problem: A Picture Frame with a 5-inch by 7-inch Photograph

When it comes to picture frames, understanding the dimensions and measurements is crucial. In this problem, we are given a 5-inch by 7-inch photograph placed inside a picture frame. The length and width of the frame are $2x$ inches larger than the width and length of the photograph. Our goal is to find the expression that represents the perimeter of the frame.

Breaking Down the Problem

To solve this problem, we need to understand the concept of perimeter and how it relates to the dimensions of the frame. The perimeter of a shape is the distance around the shape. In this case, we are dealing with a rectangular frame, so we need to find the sum of the lengths of all its sides.

Calculating the Dimensions of the Frame

Let's start by calculating the dimensions of the frame. The length of the frame is $2x$ inches larger than the length of the photograph, which is 7 inches. Therefore, the length of the frame is $7 + 2x$ inches. Similarly, the width of the frame is $2x$ inches larger than the width of the photograph, which is 5 inches. Therefore, the width of the frame is $5 + 2x$ inches.

Finding the Perimeter of the Frame

Now that we have the dimensions of the frame, we can find its perimeter. The perimeter of a rectangle is given by the formula $2 \times (\text{length} + \text{width})$. In this case, the length of the frame is $7 + 2x$ inches, and the width of the frame is $5 + 2x$ inches. Therefore, the perimeter of the frame is:

$$2 \times ((7 + 2x) + (5 + 2x))$$

Simplifying the expression, we get:

$$2 \times (12 + 4x)$$

$$24 + 8x$$

Conclusion

In conclusion, the expression that represents the perimeter of the frame is $24 + 8x$. This expression takes into account the dimensions of the frame and the fact that the length and width of the frame are $2x$ inches larger than the width and length of the photograph.

Discussion and Analysis

This problem requires a good understanding of algebraic expressions and the concept of perimeter. The solution involves breaking down the problem into smaller steps, calculating the dimensions of the frame, and finding the perimeter using the formula. The expression $24 + 8x$ represents the perimeter of the frame, which is a key concept in mathematics.

Real-World Applications

This problem has real-world applications in various fields, such as art, design, and architecture. When creating a picture frame, it's essential to consider the dimensions of the frame and the photograph to ensure a proper fit. This problem demonstrates the importance of mathematical calculations in everyday life.

Common Mistakes

When solving this problem, common mistakes include:

• Not considering the dimensions of the frame and the photograph
• Not using the correct formula for the perimeter of a rectangle
• Not simplifying the expression correctly

Tips and Tricks

To solve this problem, follow these tips and tricks:

• Break down the problem into smaller steps
• Calculate the dimensions of the frame carefully
• Use the correct formula for the perimeter of a rectangle
• Simplify the expression correctly

Conclusion

In conclusion, the expression that represents the perimeter of the frame is $24 + 8x$. This problem requires a good understanding of algebraic expressions and the concept of perimeter. The solution involves breaking down the problem into smaller steps, calculating the dimensions of the frame, and finding the perimeter using the formula.
A 5-inch by 7-inch Photograph in a Picture Frame: Q&A

In our previous article, we explored the problem of a 5-inch by 7-inch photograph placed inside a picture frame. The length and width of the frame are $2x$ inches larger than the width and length of the photograph. We found that the expression that represents the perimeter of the frame is $24 + 8x$. In this article, we'll answer some frequently asked questions related to this problem.

Q: What is the perimeter of the frame when x = 0?

A: When x = 0, the length and width of the frame are equal to the length and width of the photograph. Therefore, the perimeter of the frame is equal to the perimeter of the photograph, which is 2(7 + 5) = 24 inches.

Q: How does the perimeter of the frame change when x increases?

A: When x increases, the length and width of the frame increase by 2x inches. Therefore, the perimeter of the frame increases by 4x inches.

Q: What is the perimeter of the frame when x = 1?

A: When x = 1, the length and width of the frame are 7 + 2(1) = 9 inches and 5 + 2(1) = 7 inches, respectively. Therefore, the perimeter of the frame is 2(9 + 7) = 32 inches.

Q: How does the perimeter of the frame compare to the perimeter of the photograph?

A: The perimeter of the frame is always greater than the perimeter of the photograph. This is because the frame adds 2x inches to the length and width of the photograph, resulting in a larger perimeter.

Q: What is the relationship between the perimeter of the frame and the value of x?

A: The perimeter of the frame is directly proportional to the value of x. As x increases, the perimeter of the frame also increases.

Q: Can the perimeter of the frame be negative?

A: No, the perimeter of the frame cannot be negative. The perimeter of a shape is always a positive value.

Q: How does the problem change if the photograph is placed in a square frame?

A: If the photograph is placed in a square frame, the length and width of the frame are equal. Therefore, the perimeter of the frame is 4 times the length of the frame, which is 4(7 + 2x) = 28 + 8x inches.

Q: Can the problem be solved using a different method?

A: Yes, the problem can be solved using a different method. For example, we can use the formula for the perimeter of a rectangle, which is 2(length + width). We can also use algebraic manipulations to simplify the expression and find the perimeter of the frame.

Conclusion

In conclusion, the expression that represents the perimeter of the frame is $24 + 8x$. This problem requires a good understanding of algebraic expressions and the concept of perimeter. The solution involves breaking down the problem into smaller steps, calculating the dimensions of the frame, and finding the perimeter using the formula. We've also answered some frequently asked questions related to this problem, including how the perimeter of the frame changes when x increases and how the perimeter of the frame compares to the perimeter of the photograph.