Given that a pyramid and a cone both have the same height of 10 centimeters and they also have the same volume, let's analyze the relationship between their properties.
We know the formulas for the volumes of a pyramid and a cone are:
- Volume of a pyramid ([tex]\(V_p\)[/tex]): [tex]\( V_p = \frac{1}{3} \times \text{base area of pyramid} \times \text{height} \)[/tex]
- Volume of a cone ([tex]\(V_c\)[/tex]): [tex]\( V_c = \frac{1}{3} \times \text{base area of cone} \times \text{height} \)[/tex]
Given that the heights of the pyramid and the cone are equal, let's denote that height as [tex]\( h \)[/tex]. Thus, the height [tex]\( h = 10 \)[/tex] centimeters for both shapes.
Since the volumes of both shapes are equal, [tex]\( V_p = V_c \)[/tex].
By substituting the formulas into the equality, we get:
[tex]\[ \frac{1}{3} \times \text{base area of pyramid} \times h = \frac{1}{3} \times \text{base area of cone} \times h \][/tex]
Since the heights are the same ([tex]\( h \)[/tex]), it simplifies to:
[tex]\[ \text{base area of pyramid} = \text{base area of cone} \][/tex]
This implies that at any given height (cross-section) from the base to the apex, the cross-sectional areas of the pyramid and the cone must be equal. Hence, the horizontal cross-sections of both the pyramid and the cone at the same height must have the same area.
Thus, the correct statement that must be true about the two solids is:
OD. The horizontal cross-sections of the pyramid and cone at the same height must have the same area.
