Question-36
A rectangular paper sheet of dimensions \(54\) cm x \(4\) cm is taken. The two longer edges of the sheet are joined together to create a cylindrical tube. A cube whose surface area is equal to the area of the sheet is also taken. Then, the ratio of the volume of the cylindrical tube to the volume of the cube is
Let the length and breadth be \(\displaystyle l=54\) and \(\displaystyle b=4\). Then the radius of the cylinder is \(\displaystyle r=\frac{b}{2\pi } =\frac{2}{\pi }\) and its height is \(\displaystyle h=54\). If the side of the cube is \(\displaystyle a\), then \(\displaystyle 6a^{2} =lb\Longrightarrow a=\sqrt{\frac{lb}{6}} =6\). The ratio of the volume of the cylinder to the cube is therefore:
\[ \begin{aligned} \frac{\pi r^{2} h}{a^{3}} & =\frac{54\times 4}{\pi \times 6^{3}}\\ & \\ & =\frac{1}{\pi } \end{aligned} \]