The figure above represents a picture set in a square wooden frame that is "p" inches wide on all sides. If the combined area of picture and the frame is equal to "q" square inches, then in terms of p and q, what is the perimeter of the picture?
Show Hint
A common mistake in frame problems is to only subtract the width 'p' once. Always visualize the frame: it adds to the dimensions on two opposite sides (e.g., left AND right), so the total difference in side length between the outer and inner shapes is always 2p.
Step 1: Understanding the Concept:
We need to find the perimeter of the inner square (the picture) using the given information about the outer square (picture + frame). We must work from the outside in, starting with the total area. Step 2: Detailed Explanation:
1. Find the side length of the outer square.
The combined area of the picture and the frame is \(q\) square inches. Since the overall shape is a square, the length of one side of this outer square is the square root of its area.
\[ \text{Side length of outer square} = \sqrt{q} \]
2. Find the side length of the inner square (the picture).
The frame is \(p\) inches wide on all sides. This means to get the side length of the inner picture, we must subtract the frame's width from the outer side length. Since the frame exists on both the left and right (or top and bottom) of the picture, we must subtract the width \(p\) twice.
\[ \text{Side length of picture} = (\text{Side length of outer square}) - p - p \]
\[ \text{Side length of picture} = \sqrt{q} - 2p \]
3. Calculate the perimeter of the picture.
The picture is also a square. The perimeter of a square is 4 times its side length.
\[ \text{Perimeter of picture} = 4 \times (\text{Side length of picture}) \]
\[ \text{Perimeter of picture} = 4 \times (\sqrt{q} - 2p) \]
Distribute the 4:
\[ \text{Perimeter of picture} = 4\sqrt{q} - 8p \] Step 3: Final Answer:
The perimeter of the picture is \(4\sqrt{q} - 8p\). This corresponds to option (E).
