Let the radius of the base of both the right circular cylinder and the cone be \(r\) units. Let the height of the cylinder be \(h_1\) units and the height of the cone be \(h_2\) units.
According to the question, \[ \pi r^2 h_1 : \frac{1}{3} \pi r^2 h_2 = 3 : 2 \] Or, \[ h_1 : \frac{1}{3} h_2 = 3 : 2 \] Or, \[ \frac{h_1}{\frac{1}{3} h_2} = \frac{3}{2} \] Or, \[ \frac{h_1}{h_2} = \frac{3}{2} \times \frac{1}{3} \] Or, \[ \frac{h_1}{h_2} = \frac{1}{2} \] Therefore, \[ h_2 = 2h_1 \] ∴ The height of the cone is double the height of the cylinder. (Proved)