Multiple choice

The ratio of the total surface area of a closed cylindrical vessel to the curved surface area of the same cylindrical vessel is 4 : 3 and the height of the cylindrical vessel is 14 cm more than the radius of the cylindrical vessel. Two cylindrical vessels of this type are filled up to 75% of their capacity with water. The total water from these cylindrical vessels is transferred into a number of small cylindrical vessel whose height is 2/3 of the original cylindrical vessel and the radius is one fourth of the original cylindrical vessel. Find the number of the small cylindrical vessels.

  1. 24

  2. 28

  3. 32

  4. 36

  5. 45
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D Correct answer
Explanation

Total surface area (2*pi*r*(r+h)) to curved surface area (2*pi*r*h) is (r+h)/h = 4/3. Given h = r + 14, we get (r + r + 14)/(r + 14) = 4/3, leading to 6r + 42 = 4r + 56, so 2r = 14, r = 7, h = 21. Volume of one vessel is pi * 7^2 * 21 = 1029 * pi. Total water from two vessels is 2 * 0.75 * 1029 * pi = 1543.5 * pi. Small vessel volume is pi * (7/4)^2 * (21 * 2/3) = pi * 49/16 * 14 = 42.875 * pi. Number of vessels = 1543.5 / 42.875 = 36.

AI explanation

The ratio of total surface area to curved surface area of a cylinder is (2*pi*r*h + 2*pi*r^2) / (2*pi*r*h), which simplifies to 1 + r/h. Setting this equal to 4/3 gives r/h = 1/3, or h = 3r; since h is also r + 14, we get r = 7 cm and h = 21 cm. The total volume of water in two large cylinders is 2 * 0.75 * pi * 7^2 * 21. The volume of a small cylinder is pi * (7/4)^2 * (14), so dividing the total water volume by the small cylinder volume gives (1.5 * pi * 49 * 21) / (pi * 12.25 * 14) = 36.

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