Question

A rectangular paper sheet of dimensions \( 54 \, \text{cm} \times 4 \, \text{cm} \) 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:

• A. \( \frac{1}{\pi} \)
• B. \( \frac{2}{\pi} \)
• C. \( \frac{3}{\pi} \)
• D. \( \frac{4}{\pi} \)

Hint:

For problems involving 3D shapes and their dimensions, ensure you carefully relate the given parameters (surface area, volume, etc.) to the relevant formulas for cylinders and cubes.

Answer 1

The Correct Option is A

Step 1: Determine the area of the rectangular sheet.
The area of the rectangular sheet is given by: \[ \text{Area} = 54 \times 4 = 216 \, \text{cm}^2. \] Step 2: Dimensions of the cylindrical tube.
When the longer edges of the sheet are joined, the circumference of the cylinder's base is \( 4 \, \text{cm} \), and the height of the cylinder is \( 54 \, \text{cm} \). The radius of the base of the cylinder, \( r \), is: \[ r = \frac{\text{Circumference}}{2\pi} = \frac{4}{2\pi} = \frac{2}{\pi} \, \text{cm}. \] The volume of the cylinder is: \[ V_{\text{cylinder}} = \pi r^2 h = \pi \left( \frac{2}{\pi} \right)^2 \times 54 = \pi \cdot \frac{4}{\pi^2} \cdot 54 = \frac{216}{\pi} \, \text{cm}^3. \] Step 3: Dimensions of the cube.
The surface area of the cube is equal to the area of the sheet, which is \( 216 \, \text{cm}^2 \). For a cube, the surface area is \( 6a^2 \), where \( a \) is the side length. \[ 6a^2 = 216 \quad \Rightarrow \quad a^2 = 36 \quad \Rightarrow \quad a = 6 \, \text{cm}. \] The volume of the cube is: \[ V_{\text{cube}} = a^3 = 6^3 = 216 \, \text{cm}^3. \] Step 4: Ratio of the volumes.
The ratio of the volume of the cylindrical tube to the volume of the cube is: \[ \text{Ratio} = \frac{V_{\text{cylinder}}}{V_{\text{cube}}} = \frac{\frac{216}{\pi}}{216} = \frac{1}{\pi}. \] Conclusion: The ratio of the volume of the cylindrical tube to the volume of the cube is \( \frac{1}{\pi} \).