Question

the following figures all have the same height and same base area.
for which of the figures can we apply cavalieris principle to show they have the same volume?
choose 1 answer:
a only a and b
b only a and c
c only b and c
d a, b, and c

Explanation:

Cavalieri's principle states that if two (or more) solids have the same height and the same cross - sectional area at every level parallel to the bases, then they have the same volume. Figure A (a rectangular prism) and Figure B (a parallelepiped) are both prisms. For a prism, the volume formula is \(V = Bh\) (where \(B\) is the base area and \(h\) is the height), and at any height parallel to the base, the cross - sectional area is equal to the base area. However, Figure C is a pyramid, and the volume of a pyramid is \(V=\frac{1}{3}Bh\). The cross - sectional area of a pyramid at a height \(y\) (measured from the base) is not equal to the base area (it follows a quadratic relationship with the height). So, only the prisms (A and B) satisfy the conditions of Cavalieri's principle for having the same volume.

Answer:

A. Only A and B