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
The radius and the height
What ratio of volumes ?????
Yet the robot tutor answered it !!!!
To have an equal ratio of volumes for three solids, the measurements that would have to be equal in all three solids are the radius and the height (option b).
To determine which measurements would have to be equal in all three solids for the ratio of volumes to be true, let's consider the formula for each of these measurements in the context of different solids.
a) The volume and the height: In general, the volume of a solid depends on its base area and height. However, the base areas of three different solids may not be equal. Therefore, for the ratio of volumes to be true, the heights of the solids must be equal, but the base areas can vary.
b) The radius and the height: This measurement is specific to solids with a circular base, such as cylinders or cones. The volume of a cylinder or cone depends on the product of its base area (πr²) and height (h). For the ratio of volumes to be true, the radii and heights of the circular bases in all three solids must be equal.
c) The radius and the surface area: The surface area of a solid depends on various measurements, depending on the shape of the solid. For example, the surface area of a sphere depends on its radius, while the lateral surface area of a cylinder depends on its height and radius. Since the surface areas can vary among different solids, this option is not correct.
d) The surface area and the base: This option is not correct because the base is just a part of the entire surface area of a solid. Therefore, the surface area and the base cannot be the only measurements that need to be equal for the ratio of volumes to be true.
Based on the analysis above, option b) The radius and the height, is the correct answer. The radii and heights of the circular bases in all three solids must be equal for the ratio of volumes to be true.