Prove that the volumes of a cone, a hemisphere and a cylinder, when each has equal base radius and height, are in the proportion
\(1:2:3\).
Proof:
Let \(r\) be the radius and \(h\) be the height.
Volume of a cone \(=\)
Volume of a hemisphere \(=\)
Volume of a cylinder \(=\)
Ratio of the volumes of cone, hemisphere and cylinder \(=\) [Since \(r\) \(=\) \(h\)]
By simplifying this we get the required ratio\(=\) \(1 : 2 : 3\)
Therefore, the ratio of the volume of the cone, the hemisphere and the cylinder of equal radius and height is equal to \(1: 2 : 3\).
Answer variants:
\(\frac{2}{3} \pi r^3\)
\(πrl+πr^2 : 2πrh+2πr^2 : 3πr^2\)
\(\frac{1}{3} \pi r^2 h\)
\(πrl+πr^2\)
\(πrl\)
\(\frac{1}{3} \pi r^3\) \(:\) \(\frac{2}{3} \pi r^3\) \(:\) \(\pi r^3\)
\(\pi r^2 h\)