Suppose a sphere, cylinder, and a cone all share the same radius and the cylinder and cone share the same height, which is twice the radius. Which of the following is true about the relationship between the volume formulas for the sphere, cylinder, and cone
The volume formulas for a sphere, cylinder, and cone are as follows:
Sphere: $\frac{4}{3}\pi r^3$
Cylinder: $\pi r^2 h$
Cone: $\frac{1}{3} \pi r^2 h$
Let's assume the radius of these three objects is $r$ and the height is $h$. Given that the height of the cylinder and cone is twice the radius (i.e., $h = 2r$), we can substitute this value into the volume formulas:
Sphere: $\frac{4}{3}\pi r^3$
Cylinder: $\pi r^2 (2r) = 2\pi r^3$
Cone: $\frac{1}{3} \pi r^2 (2r) = \frac{2}{3}\pi r^3$
Comparing the volume formulas, we can see that the relationship between them is:
Sphere: $\frac{4}{3}\pi r^3$
Cylinder: $2\pi r^3 = \frac{2}{3} \times 3\pi r^3$
Cone: $\frac{2}{3}\pi r^3$
Therefore, the relationship between the volume formulas for the sphere, cylinder, and cone is that the volume of the cylinder is twice the volume of the cone, and the volume of the sphere is $\frac{3}{2}$ times the volume of the cylinder or cone.