A rectangular paper sheet of dimensions 54 cm × 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
Answer
(A)
Explanation
Joining the two 54 cm edges makes a tube 54 cm long whose circular end has a circumference of 4 cm, so its radius is \(r = \frac{4}{2\pi} = \frac{2}{\pi}\) cm and
\[V_{\text{tube}} = \pi r^{2}h = \pi \cdot \frac{4}{\pi^{2}} \cdot 54 = \frac{216}{\pi}\ \text{cm}^{3}.\]
The sheet’s area is \(54 \times 4 = 216\ \text{cm}^{2}\). A cube with surface area \(6a^{2} = 216\) has side \(a = 6\) cm and volume \(216\ \text{cm}^{3}\).
The ratio is \(\dfrac{216/\pi}{216} = \dfrac{1}{\pi}\).
Answer: A