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A rectangular paper sheet of dimensions 54 cm x 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 A. 3/pi B. 2/pi C. 3/pi D. 4/pi

GATE 2024 · General Aptitude · Geometry · medium

Answer: Ratio of cylinder volume to cube volume = 27/(2pi)

  1. Find cylinder dimensions: 2 pi r = 54 => r = 27/pi. Height h = 4 cm (shorter dimension of sheet).
  2. Find cube side from equal surface area: 6 s^2 = 54 x 4 = 216 => s^2 = 36 => s = 6 cm.
  3. Compute cylinder volume: V_cyl = pi x (27/pi)^2 x 4 = pi x (729/pi^2) x 4 = 2916/pi cm^3.
  4. Compute cube volume and find ratio: V_cube = 6^3 = 216 cm^3. Ratio = V_cyl / V_cube = (2916/pi) / 216 = 2916 / (216 pi) = 13.5/pi = 27/(2pi).