Calculating Frame Perimeter: A Math Problem

Let's dive into a fun geometry problem! We're going to figure out the perimeter of a picture frame. This problem involves a photograph and a frame, and we'll use some basic math to solve it. We will focus on the perimeter of the frame, a key concept in this problem.

Understanding the Problem: Framing the Situation

First, let's break down what we know. We have a photograph that's 5 inches wide and 7 inches long. This photo is placed inside a picture frame. The problem tells us something important about the frame: both its length and width are 2a inches larger than the corresponding dimensions of the photograph. This means the frame is bigger than the picture on all sides by the same amount. Our goal is to find an expression that represents the perimeter of the frame. This involves understanding how the frame's dimensions relate to the photo's and then using the formula for the perimeter of a rectangle.

To solve this, let's consider the dimensions of the frame. The photograph has a width of 5 inches, and the frame's width is 2a inches larger. So, the width of the frame is 5 + 2a inches. Similarly, the photograph has a length of 7 inches, and the frame's length is also 2a inches larger. Therefore, the length of the frame is 7 + 2a inches. Now, we have the dimensions of the frame in terms of a. The perimeter is the total distance around the outside of the frame, and it's essential for this problem.

Now, let's look at the concept of perimeter. The perimeter of a rectangle is calculated by adding up the lengths of all four sides. Since a rectangle has two pairs of equal sides, we can use the formula: Perimeter = 2 * (length + width). It's crucial to understand how to apply this formula to the frame's dimensions. Our next step is to substitute the frame's length and width into the perimeter formula and simplify the expression. We'll use the frame's calculated length (7 + 2a) and width (5 + 2a) in the formula. This step will bring us closer to finding the final expression that represents the perimeter of the frame. The perimeter is a fundamental concept in geometry, and correctly calculating it is key to the solution.

Breaking Down the Dimensions: From Photo to Frame

Let's get into the specifics. The photograph is 5 inches by 7 inches. The problem gives us the critical information that both the length and width of the frame are 2a inches larger than the photo. This means we must add 2a to both the length and the width of the photograph to find the frame's dimensions. Visualizing this is helpful: imagine the photo nestled snugly inside the frame, with the frame extending out by the same amount on all sides. This consistent extension is what makes the problem solvable using a single variable, a. This consistent extension is a key aspect for solving the problem.

So, the frame's width is the photo's width (5 inches) plus 2a: 5 + 2a. The frame's length is the photo's length (7 inches) plus 2a: 7 + 2a. Now we have both the width and length of the frame expressed in terms of a. It's critical to note how each dimension of the frame is determined by the value of a. As a changes, the frame's size changes accordingly. It highlights the direct relationship between the frame's dimensions and the variable. Therefore, understanding this relationship is key to the following steps.

Now we've got all the pieces we need: the original dimensions of the photo and the critical relationship of 2a inches increase in both length and width to find the frame dimensions. The problem is now essentially a word problem that translates into a mathematical equation. Understanding the frame's dimensions is the core of this problem.

Calculating the Perimeter: Putting It All Together

Now for the main event: calculating the perimeter of the frame. We know the frame is a rectangle. Also, we know the perimeter of a rectangle is found using the formula: Perimeter = 2 * (length + width). We've already worked out the frame's length and width in the previous steps. The frame's length is (7 + 2a) inches, and its width is (5 + 2a) inches. Now we'll substitute these values into the perimeter formula, following the established formula structure.

So, the perimeter of the frame is 2 * ((7 + 2a) + (5 + 2a)). Let's simplify this. First, combine the terms inside the parentheses: 7 + 5 = 12, and 2a + 2a = 4a. So, the expression inside the parentheses simplifies to 12 + 4a. Next, multiply the entire expression by 2: 2 * (12 + 4a) = 24 + 8a. Therefore, the expression that represents the perimeter of the frame is 24 + 8a inches. This final result represents the total distance around the outside of the frame, accounting for the photo's size and the extra width added by the frame itself. The simplified formula, Perimeter = 24 + 8*a is the answer to the problem.

This is a great example of how to solve a practical geometry problem. We started with some basic information, broke down the problem into smaller parts, and applied the correct formulas to find a solution. The key was understanding how the frame's dimensions related to the photo's dimensions, then skillfully applying the perimeter formula. By understanding each step, anyone can solve similar problems, improving their understanding of geometric concepts. The perimeter of the frame is represented by this equation.

Simplifying the Expression: From Formula to Answer

Let's go through the simplification step by step to ensure we understand how the final expression is derived. We have the formula: Perimeter = 2 * ((7 + 2a) + (5 + 2a)). Our goal is to simplify this expression to its simplest form. The first step involves combining like terms within the parentheses. We have two sets of terms inside the parentheses: (7 + 2a) and (5 + 2a). Combining the constants (7 and 5) gives us 12. Combining the terms with a (2a and 2a) gives us 4a. So, the expression inside the parentheses simplifies to 12 + 4a. This step simplifies the entire process.

Now, our formula looks like this: Perimeter = 2 * (12 + 4a). The next step is to distribute the 2 across the terms inside the parentheses. We multiply both 12 and 4a by 2. 2 multiplied by 12 equals 24, and 2 multiplied by 4a equals 8a. This distribution step is very important. Therefore, our simplified expression for the perimeter is 24 + 8a. This is the final and simplified answer, representing the total distance around the outside of the frame in terms of a. The simplified expression is the final answer, ensuring accuracy and clarity. The perimeter of the frame is solved.

Real-World Applications: Where Frames and Math Meet

This isn't just an abstract math problem. It has real-world applications! Think about it: Picture framing, wallpapering, and even construction projects often involve calculating perimeters and areas. Understanding how to calculate the perimeter is incredibly useful in various practical scenarios. For instance, when you're framing a picture, you might need to determine how much framing material to purchase. This calculation directly uses the perimeter formula we just practiced. You would know the dimensions of the photo and the desired size of the frame, much like in our problem. These skills are valuable in many areas.

Consider wallpapering a room. You must calculate the perimeter to figure out the total length of the room's walls. This is a very practical use of the perimeter calculation. The same concept is relevant in construction when calculating the length of fencing needed for a property. These examples showcase the importance of understanding the concepts. It is not limited to math class. Being able to solve such problems makes you able to make informed decisions in these situations.

Understanding perimeter also helps with understanding the concept of area. The area is the amount of space inside a shape, while the perimeter is the distance around the outside. These two concepts often go hand-in-hand in practical applications. The knowledge is useful whether you're decorating your home or planning a major construction project. The applications are broad and diverse, highlighting the importance of the concepts.

Conclusion: Framing Your Understanding

We've successfully calculated the perimeter of a picture frame, turning a word problem into a clear mathematical expression. We started with the dimensions of a photograph and the information about how the frame relates to the photo. Then, we used the perimeter formula for a rectangle to find our answer. The ability to break down the problem step-by-step is an important skill. The ability to simplify mathematical expressions and apply formulas correctly is a key part of solving this problem. Congratulations! You've successfully navigated a geometry problem, understood the concepts of perimeter and dimensions. This exercise helps to reinforce the mathematical concepts, encouraging continued exploration and enhancing practical problem-solving skills.

For more information, consider exploring resources on geometry and problem-solving. This will help you deepen your understanding and apply these skills in different contexts.

If you would like to explore more about perimeter and geometry, here is a link to Khan Academy.