Question

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 1

$\frac{27}{2\pi}$

Answer 2

1. Calculate the dimensions of the cylinder formed by rolling the sheet: the longer edge (54 cm) becomes the circumference, and the shorter edge (4 cm) becomes the height. Use the circumference formula $2\pi r$ to find the radius $r$.
2. Calculate the volume of the cylinder using the formula $V = \pi r^2 h$.
3. Calculate the area of the rectangular sheet.
4. The surface area of the cube is equal to the area of the sheet. Use the surface area formula of a cube $6a^2$ to find the side length $a$ of the cube.
5. Calculate the volume of the cube using the formula $V = a^3$.
6. Find the ratio of the cylinder's volume to the cube's volume and simplify the expression.