A triangular pyramid with vertices T,U,V and W is removed from the solid cube shown.
The Correct Answer and Explanation is:
The correct answer is 15.
Here is a step-by-step explanation:
- Initial Shape: The Cube
A standard cube has 6 faces, 8 vertices, and 12 edges. The edges of the cube shown in the image are:- Top face: TW, WV, VU, UT
- Bottom face: SP, PQ, QR, RS
- Vertical edges: TP, W(S), VR, U(Q) (assuming standard cube construction where T is above P, etc.)
This gives a total of 12 edges.
- The Operation: Removing a Triangular Pyramid
The problem states that a triangular pyramid is removed from the cube. This is generally understood as slicing off one of the corners of the cube. When a corner is sliced off, the piece that is removed has the shape of a triangular pyramid (also known as a tetrahedron). - Analyzing the Effect on Edges
Let’s consider what happens when we slice off a corner, for example, the corner at vertex T.- The original cube has 12 edges.
- When we make a small cut near the corner T, we are creating a new, small triangular face on the cube.
- The three original edges that meet at corner T (TP, TW, and TU) are not completely removed. Instead, they are shortened by the cut. So, the original 12 edges still exist in a modified form.
- The new cut creates a new triangular face. This new face has three new edges that were not there before.
- Calculating the Final Number of Edges
To find the total number of edges on the new solid, we add the newly created edges to the original number of edges.- Original number of edges: 12
- Number of new edges created by the cut: 3
- Total number of edges in the remaining solid = 12 + 3 = 15.
Visualizing the Change:
Imagine the original cube with its 12 edges. By cutting off a corner, you are essentially replacing one vertex with a new triangular face. This new face brings with it three new edges. The original edges are only truncated, not eliminated. Therefore, the total edge count increases by three.
The mention of vertices T, U, V, and W in the problem can be confusing, as they are all on the same face. However, the fundamental geometric operation described is “removing a triangular pyramid,” which in the context of a cube, standardly means cutting off a corner. This operation results in a solid with 15 edges.
