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. \(1/\pi\)
• B. \(2/\pi\)
• C. \(3/\pi\)
• D. \(4/\pi\)

Hint:

"Join longer edges" \(\Rightarrow \) shorter side becomes the circumference. Equate sheet area directly with \(6a^2\) to get the cube edge quickly.

Answer 1

The Correct Option is A

Step 1: Form the cylinder from the sheet.
Joining the longer edges (\(54\text{ cm}\)) means the shorter side \(4\text{ cm}\) becomes the circumference of the circular base, and the height equals \(54\text{ cm}\).
Circumference \(= 2\pi r = 4 \Rightarrow r = \dfrac{2}{\pi}\) cm, height \(h = 54\) cm.
Step 2: Volume of the cylindrical tube.
\(V_{\text{cyl}} = \pi r^2 h = \pi\left(\dfrac{2}{\pi}\right)^2 \cdot 54 = \pi \cdot \dfrac{4}{\pi^2} \cdot 54 = \dfrac{216}{\pi}\;\text{cm}^3.\)
Step 3: Cube whose surface area equals the sheet's area.
Area of sheet \(= 54 \times 4 = 216\;\text{cm}^2\).
Let cube edge be \(a\). Surface area \(= 6a^2 = 216 \Rightarrow a^2 = 36 \Rightarrow a = 6\) cm.
So \(V_{\text{cube}} = a^3 = 6^3 = 216\;\text{cm}^3.\)
Step 4: Required ratio.
\(\displaystyle \frac{V_{\text{cyl}}}{V_{\text{cube}}} = \frac{216/\pi}{216} = \boxed{\frac{1}{\pi}}.\)