Correct option is: B) 42 cm
Let radius and height of the cone be r and h respectively.
Given that vertical angle of cone is \(120^\circ\).
i.e \(\angle CBA = 120^\circ\)
\(\Rightarrow\) \(\angle ABO = \frac {\angle CBA}{2} = \frac {120^\circ}{2} = 60^\circ\)
In right \(\triangle AOB\),
\(\frac {OA}{OB} = tan \, 60^\circ = \sqrt3\)
\(\Rightarrow\) \(\frac rh = \sqrt3\)
\(\Rightarrow\) r = \(h\sqrt3\)
Volume of cone = volume of spherical ball
\(\Rightarrow\) \(\frac 13 \pi r^2h\) = 232848
\(\Rightarrow\) \(\frac 13 \pi \times 3h^3\) = 232848
\(\Rightarrow\) \(h^3 = \frac {232848}\pi= \frac {232848\times7}{22}\) = 10584 \(\times\) 7 = 74088 = \(42^3\)
\(\therefore\) h = 42 cm.
Hence, the height of the cone is 42 cm.
