A right circular cone of radius 7 cm and height 12 cm is completely filled with water. This water is then poured into an empty cylindrical vessel of radius 3.5 cm. What will be the height (in centimetres) of the water column in the cylinder?

Introduction / Context:
This problem involves the conservation of volume when liquid is transferred between two containers of different shapes. A right circular cone is filled with water, and then all this water is poured into a right circular cylinder. Since the water is neither lost nor added, the volume of water in the cone equals the volume of water in the cylinder. Using the formulas for the volumes of a cone and a cylinder, we can determine the height of the water in the cylindrical vessel.

Given Data / Assumptions:
- Cone radius r_cone = 7 cm.- Cone height h_cone = 12 cm.- Cylinder radius r_cyl = 3.5 cm.- All water from the cone is poured into the cylinder.- There is no spillage or evaporation; volume of water remains constant.- π is the same for both shapes in the calculations.

Concept / Approach:
Volume of a right circular cone is given by V_cone = (1 / 3) * π * r_cone^2 * h_cone. Volume of a right circular cylinder is given by V_cyl = π * r_cyl^2 * h_cyl, where h_cyl is the height of water in the cylinder that we need to find. Since the water volumes are equal, we set V_cone = V_cyl and solve for h_cyl. This uses algebraic manipulation and simple squaring of the radii to find the unknown height.

Step-by-Step Solution:
Step 1: Use the cone volume formula: V_cone = (1 / 3) * π * r_cone^2 * h_cone.Step 2: Substitute r_cone = 7 and h_cone = 12.Step 3: Compute r_cone^2 = 7^2 = 49.Step 4: So V_cone = (1 / 3) * π * 49 * 12.Step 5: 49 * 12 = 588, so V_cone = (1 / 3) * 588 * π = 196 * π.Step 6: For the cylinder, volume V_cyl = π * r_cyl^2 * h_cyl.Step 7: r_cyl = 3.5 cm, so r_cyl^2 = 3.5^2 = 12.25.Step 8: Therefore, V_cyl = π * 12.25 * h_cyl.Step 9: Set V_cone = V_cyl: 196 * π = π * 12.25 * h_cyl.Step 10: Cancel π from both sides to get 196 = 12.25 * h_cyl.Step 11: Solve for h_cyl: h_cyl = 196 / 12.25 = 16 cm.

Verification / Alternative check:
We can check computations numerically. 12.25 * 16 = 196, confirming that the height h_cyl of 16 cm is consistent with the equality of volumes. Another way is to see that the radius of the cylinder is exactly half the radius of the cone (7 and 3.5), so the base area of the cylinder is one fourth that of the cone. Since the cone volume is one third of a cylinder with the same base and height, adjusting for the changed radius shows that to hold the same volume the cylinder must have a proportionally higher height, which matches 16 cm in this configuration.

Why Other Options Are Wrong:
- 32 cm: Doubling the correct height would double the volume beyond what is available from the cone.- 5.33 cm: This is far too small; substituting this height would yield a volume much less than 196π.- 8 cm: This is half of the true height and cannot hold the full cone volume at the given radius.- 12 cm: Although it matches the height of the cone, the cylinder has a smaller radius, so the same height would not produce the same volume.

Common Pitfalls:
Learners sometimes incorrectly use the same radius for both shapes or forget the 1 / 3 factor in the volume of a cone, leading to incorrect equalities. Others may accidentally square the height instead of the radius. Also, forgetting to cancel π from both sides complicates the algebra unnecessarily. Recognising that the radii differ and that the cone has the additional one third factor is key to setting up the correct equation.

Final Answer:
The height of water in the cylindrical vessel is 16 cm.