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:

When converting a rectangle to a cylinder, the length forms the circumference, and the height remains constant.

Answer 1

The Correct Option is A

The problem involves comparing the volumes of a cube and a cylinder derived from the same surface area. Step 1: Surface area of the sheet \[ \text{Area of sheet} = 54 \times 4 = 216 \, \text{cm}^2 \] This area is given as the surface area of the cube. Step 2: Side of the cube Let the side of the cube be \(a\). The surface area of a cube is: \[ 6a^2 = 216 \] \[ a^2 = \frac{216}{6} = 36 \quad \implies \quad a = 6 \, \text{cm}. \] Hence, the volume of the cube is: \[ \text{Volume of cube} = a^3 = 6^3 = 216 \, \text{cm}^3. \] Step 3: Radius of the cylinder The same surface area is used to form a cylinder. The surface area of the cylinder includes the lateral surface area and the areas of the two circular bases: \[ \text{Circumference of top and bottom circles} = 4 \quad \text{(from the lateral surface)}. \] The circumference is given by: \[ 2 \pi R_c = 4 \quad \implies \quad R_c = \frac{2}{\pi}. \] Step 4: Volume of the cylinder The volume of the cylinder is: \[ \text{Volume of cylinder} = \pi R_c^2 h, \] where \(h = 54 \, \text{cm}\) (height of the cylinder). Substituting \(R_c = \frac{2}{\pi}\): \[ \text{Volume of cylinder} = \pi \left( \frac{2}{\pi} \right)^2 \times 54 = \pi \cdot \frac{4}{\pi^2} \cdot 54 = \frac{4 \cdot 54}{\pi} = \frac{216}{\pi} \, \text{cm}^3. \] Step 5: Ratio of volumes The ratio of the volumes of the cylinder to the cube is: \[ \text{Ratio} = \frac{\text{Volume of cylinder}}{\text{Volume of cube}} = \frac{\frac{216}{\pi}}{216} = \frac{1}{\pi}. \] Final Answer: \[ \boxed{\frac{1}{\pi}} \]