Suppose a sphere, cylinder, and cone all share the same radius and the cylinder and cone also share the same height, which is twice the radius. Explain how 3 cones = 1 cylinder.
To understand how three cones can be equal to one cylinder, we need to first examine their respective volumes.
The volume of a sphere is given by the formula:
\[V_{\text{sphere}} = \frac{4}{3} \pi r^3\]
The volume of a cylinder is given by the formula:
\[V_{\text{cylinder}} = \pi r^2 h\]
The volume of a cone is given by the formula:
\[V_{\text{cone}} = \frac{1}{3} \pi r^2 h\]
Given that the radius of the sphere, cylinder, and cone is the same, let's denote it as \(r\). The height of the cylinder and the cone is twice the radius, so we denote it as \(h = 2r\).
The volume of the sphere is then:
\[V_{\text{sphere}} = \frac{4}{3} \pi r^3\]
The volume of the cylinder is:
\[V_{\text{cylinder}} = \pi r^2 (2r) = 2 \pi r^3\]
And the volume of the cone is:
\[V_{\text{cone}} = \frac{1}{3} \pi r^2 (2r) = \frac{2}{3} \pi r^3\]
Now, if we compare the volumes of three cones to one cylinder, we have:
\[3 \times V_{\text{cone}} = 3 \times \left(\frac{2}{3} \pi r^3\right) = 2 \pi r^3\]
This is equal to the volume of the cylinder:
\[V_{\text{cylinder}} = 2 \pi r^3\]
Therefore, we can conclude that the volume of three cones is equal to the volume of one cylinder when they all have the same radius and the cylinder and cone also have the same height (twice the radius).