question_answer

From a solid wooden right circular cylinder, a right circular cone whose radius and height are same as the radius and height of the cylinder respectively, is curved out. What is the ratio of the volume of the utilised wood to that of the wasted wood?

A) 1 : 2     

B) 2 : 1

C) 2 : 3                             

D) 1 : 3

Correct Answer: A

Solution :

Let the height and radius of right circular cylinder are h and r, respectively.

Then, volume or cylinder \[\pi {{r}^{2}}h\]

By condition,

volume of circular cone \[=\frac{1}{3}\pi {{r}^{2}}h\]

\[\therefore \] Required ratio

\[\text{=}\,\,\frac{\text{Volume}\,\text{of}\,\text{wasted}\,\text{wood}}{\text{Volume}\,\text{of}\,\text{utilised}\,\text{wood}}\]

\[\text{=}\,\,\,\,\frac{\text{Volume}\,\text{of}\,\text{right}\,\text{circular}\,\text{cone}}{\left[ \begin{align}   & \text{Volume}\,\text{of}\,\text{right}\,\text{circular}\,\text{cylinder}\, \\  & -\text{Volume}\,\text{of}\,\text{right}\,\text{circulars}\,\text{cone} \\ \end{align} \right]}\]

\[\text{=}\,\,\frac{\frac{1}{3}\pi {{r}^{2}}h}{\left( \pi {{r}^{2}}h-\frac{1}{3}\pi {{r}^{2}}h \right)}=\,\,\frac{\frac{1}{3}\pi {{r}^{2}}h}{\frac{2}{3}\pi {{r}^{2}}h}=1:2\]