Given that a pyramid and a cone are both 10 centimeters tall and have the same volume, we want to determine which statement must be true about the two solids.
Let's analyze the properties of the pyramid and cone:
1. Volume Formula:
- For a pyramid: [tex]\( V = \frac{1}{3} \text{Base Area} \times \text{Height} \)[/tex]
- For a cone: [tex]\( V = \frac{1}{3} \pi \text{Radius}^2 \times \text{Height} \)[/tex]
Both the pyramid and the cone have the same volume and height. This ensures a relationship between their base areas (pyramid) and the base area formed by the radius (cone).
2. Cross-sections:
- A horizontal cross-section at a certain height in both the pyramid and cone is a slice parallel to the base:
- For the pyramid, this cross-section will geometrically shrink in size but retain the shape of the base.
- For the cone, this cross-section will be a circle whose radius decreases linearly with height from the base to the apex.
We know that the volumes are equal, so the areas of these sections across both shapes follow a certain pattern proportional to the height from the base.
Thus, the areas of the horizontal cross-sections of both solids, despite being different shapes (polygon for pyramid, circle for cone), will have equal areas when taken at the same height.
Therefore, the correct statement must be:
A. The horizontal cross-sections of the pyramid and cone at the same height must have the same area.
