question_answer

Find the maximum volume of a cone that can be carved out of a solid hemisphere of radius r.

A)  \[\frac{4}{3}\,\pi {{r}^{3}}\,\]                   

B)  \[2\,\pi {{r}^{3}}\,\]

C)  \[\frac{1}{3}\,\pi {{r}^{3}}\,\]        

D)         \[\frac{2}{3}\,\pi {{r}^{3}}\,\]                   

Correct Answer: C

Solution :

To cut out maximum area, radius of the cone = radius of the sphere = r  and height of the cone = radius of the cone = r            \[\therefore \] Volume of the cone \[=\frac{1}{3}\pi {{r}^{2}}h=\frac{1}{3}\pi {{r}^{3}}\]