- Identify the photograph dimensions as \(5\) and \(7\).
- Define frame dimensions as \(5 + 2a\) and \(7 + 2a\).
- Apply the perimeter formula \(P = 2(L + W)\) and substitute the frame's dimensions.
- Simplify the resulting expression to get \(P = 24 + 8a\), which is \(\boxed{8a + 24}\).
Explanation
- Analyze the dimensions of the photograph and frame
The photograph has dimensions \(5\) inches and \(7\) inches. We are told that the frame's dimensions are \(2a\) inches larger than those of the photograph. Let \(W = 5\) and \(L = 7\) be the width and length of the photograph, respectively. Then, the dimensions of the frame are \(W_f = 5 + 2a\) and \(L_f = 7 + 2a\).
- Calculate the perimeter of the frame
The perimeter \(P\) of a rectangle with length \(L_f\) and width \(W_f\) is given by the formula:
\[P = 2(L_f + W_f)\]
Substituting our expressions for the frame's dimensions:
\[P = 2((7 + 2a) + (5 + 2a))\]
Simplify the expression inside the parentheses:
\[(7 + 5) + (2a + 2a) = 12 + 4a\]
Now, distribute the factor of \(2\):
\[P = 2(12 + 4a) = 24 + 8a\]
Thus, the expression for the perimeter of the frame is \(8a + 24\).
- Conclusion
Comparing our result to the given options, we find that the expression \(8a + 24\) corresponds to the second option. Therefore, the correct expression for the perimeter of the frame is \(\boxed{8a + 24}\).
Examples
In interior design, understanding dimensions is crucial for framing artwork or photos. If you know the dimensions of your artwork and want to add a uniform frame, you can use algebra to calculate how much moulding or framing material you need. For example, if you decide on a frame width of 2 inches (meaning 2a = 2, so a = 1) for a 5x7 inch photo, you can easily calculate the required perimeter to ensure you buy enough materials for the frame. This algebraic skill helps in planning DIY projects or ordering custom sizes accurately.