Question

question 5 of 39
a pyramid and a cone are both 10 centimeters tall and have the same
volume. what statement must be true about the two solids?
a. the vertical cross - sections of the pyramid and cone at the same
width must have the same area.
b. the cross - sections of the pyramid and cone are the same shape.
c. the horizontal cross - sections of the pyramid and cone at the
same height must have the same area.
d. the area of the cross - sections of the pyramid and cone are
multiples of each other.

Explanation:

Step1: Recall the volume formula

The volume formula for a pyramid is \(V=\frac{1}{3}Bh\) (where \(B\) is the base area and \(h\) is the height), and for a cone is \(V = \frac{1}{3}\pi r^{2}h\). Given \(h_{pyramid}=h_{cone} = 10\) cm and \(V_{pyramid}=V_{cone}\).

Step2: Use Cavalieri's principle

Cavalieri's principle states that if two solids of the same height have equal - area horizontal cross - sections at every level, then they have the same volume. Since the pyramid and the cone have the same volume and the same height, the horizontal cross - sections of the pyramid and cone at the same height must have the same area.

• Option A: The vertical cross - sections of a pyramid (which can be a triangle depending on the base) and a cone (a triangle) at the same width do not necessarily have the same area. The base of the pyramid (could be a square, rectangle etc.) and the base of the cone (a circle) are different, so the relationship for vertical cross - sections is not straightforward.
• Option B: The cross - sections of a pyramid (e.g., a square - based pyramid has square or triangular cross - sections depending on the cut) and a cone (circular or triangular cross - sections) are not the same shape.
• Option D: There is no basis for the cross - sectional areas being multiples of each other. The key relationship is given by the volume formula and Cavalieri's principle which relates to equal - area horizontal cross - sections at the same height.

Answer:

C. The horizontal cross - sections of the pyramid and cone at the same height must have the same area.