Final Answer:
Tripling the radius of a pyramid or cone will increase its volume by a factor of 27.
Step-by-step explanation:
When considering the volume of a pyramid or cone, the formula involves the radius cubed (V = (1/3)πr^2h for a cone). Therefore, any change in the radius has a cubed effect on the volume. In this case, tripling the radius means multiplying it by 3, and raising that to the power of 3 (3^3), resulting in a factor of 27. This means the new volume will be 27 times greater than the original volume.
To visualize this, imagine a cone with a certain radius and volume. If you triple the radius while keeping the height constant, the base area increases ninefold (3^2), but since the height remains the same, the overall volume increases 27 times (3^3). This concept is crucial for understanding the proportional relationship between the radius and volume in pyramids and cones, emphasizing the significant impact a change in the radius has on their overall size.