In order for this ratio of volumes to be true, what measurements would have to be equal in all 3 solids?
a
The volume and the height
b
The radius and the height
c
The radius and the surface area
d
The surface area and the base
area
Answer: The radius and the height
To determine what measurements would have to be equal in all three solids for the ratio of volumes to be true, we need to understand the relationship between volume and the different measurements of solids.
The volume of a solid depends on its dimensions, which can vary depending on the shape of the solid. The measurements that affect the volume of different types of solids are:
1. For a cylinder: The volume of a cylinder is calculated using the formula V = πr^2h, where r is the radius of the base and h is the height of the cylinder.
2. For a sphere: The volume of a sphere is calculated using the formula V = (4/3)πr^3, where r is the radius of the sphere.
3. For a prism or cube: The volume of a prism or cube is calculated by multiplying the area of the base by the height.
Given this information, we can discard options b, c, and d because the measurements mentioned in those options do not directly or consistently relate to the volume of all three solids.
Therefore, the correct answer is option a: The volume and the height. If the volume and the height are equal for all three solids, the ratio of their volumes could be true.