Question

consider the following right pyramid and right prism. both figures have a base that is a regular pentagon.
heidi tried to use cavalieri’s principle to show that the two figures have the same volume.
\the base areas are the same. therefore, corresponding cross - sections have the same area. the figures have the same height. so the volumes must be the same.\
what is the first mistake heidi made?
choose 1 answer:
a the base areas are not the same.
b it isn’t true that corresponding cross - sections have the same area.
c the heights are not the same.

Explanation:

• For a right pyramid (volume formula \( V_{pyramid}=\frac{1}{3}Bh \)) and a right prism (volume formula \( V_{prism}=Bh \)) with the same base area \( B \) and height \( h \), their cross - sections at the same height:
• The cross - section of the prism at any height parallel to the base is always equal to the base area \( B \).
• The cross - section of the pyramid at a height \( y \) from the base (using similar figures) has an area that is proportional to \( (1 - \frac{y}{h})^2B \) (for a pyramid with height \( h \)). So, the cross - sectional areas of the pyramid and the prism at corresponding heights are not the same.
• Option A is incorrect because the problem states that both figures have a base that is a regular pentagon, and if we assume the bases are congruent (since they are both regular pentagons and the figures are drawn with the same base - related dimensions implied), the base areas would be the same. Option C is incorrect as the heights are both 2 (from the diagram), so they are the same.

Answer:

B. It isn't true that corresponding cross - sections have the same area.