The question asks for the shape of the outer boundary of a cross-section formed by a cutting plane passing through the vertices of two identical, inverted right circular cones joined at their common base.
Double Cone Visualization
Consider two identical right circular cones joined at their bases. One cone is upright, and the other is inverted, forming a symmetrical double cone shape. The line connecting their vertices ($V_1$ and $V_2$) is the axis of symmetry.
Cutting Plane Analysis
The cutting plane passes through both vertices ($V_1$ and $V_2$). This implies the plane contains the axis ($V_1V_2$) of the double cone.
Cross-Section Geometry
A plane containing the axis of a cone intersects it along two generators passing through the vertex.
- The plane cuts the first cone along two generators ($V_1P$ and $V_1Q$).
- It cuts the second cone along two generators ($V_2P$ and $V_2Q$).
- The plane intersects the common circular base along a diameter ($PQ$).
Rhombus Shape Formation
The outer boundary of the cross-section is formed by the four segments: $V_1P$, $V_2P$, $V_1Q$, and $V_2Q$. This creates a quadrilateral $V_1PV_2Q$.
Because the cones are identical right circular cones:
- The lengths of all generators are equal to the slant height, $L$. Thus, $V_1P = V_1Q = V_2P = V_2Q = L$.
- The quadrilateral $V_1PV_2Q$ has four sides of equal length $L$.
A quadrilateral with four equal sides is defined as a rhombus.
Final Shape Conclusion
Therefore, the outer boundary of the cross-section is a rhombus.
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The piece of paper that can be folded to make this dice is
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What is the maximum number of squares without a "hole in the interior" that can be formed within the 4 × 4 grid using the unit squares as building blocks?
