Which option will fold into the pack shown in the image below?
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When solving net-folding problems with patterns, focus on a specific, distinctive part of the pattern on the 3D object (like a corner or a 'Z' shape). Then, check each net to see if it can reproduce that specific feature when folded.
Step 1: Understanding the Concept:
This is a spatial visualization problem where we need to match a 2D net (unfolded pattern) to its corresponding 3D folded form. The key is to trace the path of the continuous red line on the 3D object and see which net reproduces that path correctly.
Step 2: Key Formula or Approach:
1. Analyze the 3D object. It consists of two identical triangular prisms joined at their square bases. Note that the object is shown from two different viewpoints.
2. Trace the red line on the 3D object. The line seems to wrap around the entire object continuously. Note where the line crosses edges between faces.
3. Mentally fold each of the 2D nets and trace the path of the red line. The correct net will form the target 3D shape, and the line segments will connect seamlessly across the folds to form the continuous path seen on the object.
Step 3: Detailed Explanation:
Let's analyze the path of the red line on the 3D shape.
The line travels across both triangular faces of one prism.
It crosses the rectangular faces.
The path is continuous onto the second prism.
Let's focus on a single prism. On the top prism shown in the left image, the line forms a 'Z' shape on the front two rectangular faces and then goes onto the triangular face.
Now, let's examine the nets:
Net (A): This net can fold into the double-prism shape. However, let's trace the line. If we fold this, the lines on the adjacent rectangular faces do not connect to form the 'Z' shape seen on the 3D model.
Net (B): The topology of this net is incorrect. It has too many faces in a row and will not fold into the compact double-prism shape. The line segments would not connect correctly.
Net (C): This net has the correct arrangement of faces to form the double-prism. Let's trace the line. We can identify the two central squares that form the base where the prisms join. The faces extending from them form the prisms. Let's focus on the right half. The two rectangular faces and the triangular face have line segments that, when folded, will create the exact pattern seen on the 3D object. The line segments on the edges will meet perfectly. For example, the segment on the rightmost rectangular face will meet the segment on the adjacent triangular face. The same logic applies to the left half of the net. The line correctly continues across the central fold. This net is correct.
Net (D): Similar to (B), the layout of the faces is a long chain, which makes it unlikely to fold into the correct shape. If we try to fold it, the line segments will not align to create the continuous path shown.
Step 4: Final Answer:
By carefully analyzing the connectivity of the faces and the continuity of the red line, we can determine that only Net (C) will correctly fold into the 3D package shown.
