A right circular cone of radius 3.5 cm and height 27 cm is completely filled with water. This water is then poured into an empty vertical cylindrical vessel of radius 2.1 cm. Assuming no water is spilled, what will be the height, in centimetres, of the water column in the cylindrical vessel? (Take pi = 22/7.)

Introduction / Context:
This question is a classic example of volume conservation when a liquid is transferred from one container to another. It involves the volumes of a cone and a cylinder, and requires you to apply the correct formulas and equate the volumes, since the quantity of water remains the same in both shapes.

Given Data / Assumptions:

• Cone radius r = 3.5 cm.

• Cone height h = 27 cm.

• Cylinder radius R = 2.1 cm.

• The cone is completely filled with water and all this water is transferred to the cylinder.

• There is no loss of water during transfer.

• Use pi = 22/7.

Concept / Approach:
The fundamental idea is that volume of water in the cone equals volume of water in the cylinder after transfer. The volume of a right circular cone is (1 / 3) * pi * r^2 * h. The volume of a right circular cylinder is pi * R^2 * H, where H is the height of the water column that we need to find. By equating these two expressions and simplifying, we can solve directly for H.

Step-by-Step Solution:
Volume of water in cone = (1 / 3) * pi * r^2 * h. Substitute r = 3.5 cm and h = 27 cm. Volume = (1 / 3) * pi * (3.5)^2 * 27. Volume of water in cylinder = pi * R^2 * H where R = 2.1 cm and H is unknown. So, (1 / 3) * pi * (3.5)^2 * 27 = pi * (2.1)^2 * H. Cancel pi from both sides: (1 / 3) * (3.5)^2 * 27 = (2.1)^2 * H. Compute left side: (3.5)^2 = 12.25, so (1 / 3) * 12.25 * 27. (1 / 3) * 27 = 9, so left side = 12.25 * 9 = 110.25. Compute (2.1)^2 = 4.41. Thus 110.25 = 4.41 * H. H = 110.25 / 4.41 = 25 cm.

Verification / Alternative check:
We can approximate to confirm: if we round 3.5 as 3.5 and 2.1 as about 2.1, the ratio of areas r^2 / R^2 becomes (3.5^2) / (2.1^2) which is roughly (12.25) / (4.41) close to 2.78. Multiplying the cone height of 27 cm by (1 / 3) and then by this ratio gives 9 * 2.78 ≈ 25, which is consistent with the exact answer. This quick mental check supports the detailed calculation.

Why Other Options Are Wrong:
The value 12.5 cm would result from mistakenly dividing by 8.82 instead of 4.41 or from misplacing the factor 1 / 3. The values 37.5 cm and 50 cm are too large and would imply that the volume in the cylinder is greater than the original cone volume, which is impossible since the amount of water is conserved. Only 25 cm satisfies the exact equality between the cone and cylinder volumes.

Common Pitfalls:
A common mistake is to forget to square the radii when applying the formulas, writing pi * r * h instead of pi * r^2 * h. Another error is to forget the factor 1 / 3 in the cone volume formula. Students may also incorrectly convert units or unnecessarily use pi = 22/7 when it cancels out anyway, leading to extra arithmetic errors. Carefully writing and simplifying the equation step by step helps avoid these issues.

Final Answer:
The height of the water column in the cylindrical vessel is 25 cm.