A solid cone of radius 3.5 centimetres and height 12 centimetres is completely filled with water and emptied into a cylindrical vessel of radius 7 centimetres. Find the height of water in the cylindrical vessel, in centimetres.

Introduction / Context:
This question involves volume conservation when liquid is transferred from one container to another. The cone is initially full of water, and that water is poured into a cylinder. Because the water volume is preserved, the volume of the cone equals the volume of water in the cylinder. Such problems are common in aptitude tests involving mensuration and practical applications.

Given Data / Assumptions:

• Radius of cone r_c = 3.5 cm.
• Height of cone h_c = 12 cm.
• Radius of cylinder r_y = 7 cm.
• Height of water in cylinder h_y is to be found.
• Liquid volume is conserved; no loss or gain.
• Use pi as a common factor that will cancel out.

Concept / Approach:
Volume of a cone:
V_cone = (1 / 3) * pi * r_c^2 * h_c.
Volume of a cylinder:
V_cylinder = pi * r_y^2 * h_y.
Since all the water from the cone is poured into the cylinder, we set V_cone = V_cylinder and solve for h_y. Because pi appears on both sides, it will cancel out, simplifying the computation.

Step-by-Step Solution:
Step 1: Write cone volume: V_cone = (1 / 3) * pi * r_c^2 * h_c. Step 2: Substitute r_c = 3.5 and h_c = 12: V_cone = (1 / 3) * pi * (3.5)^2 * 12. Step 3: Compute (3.5)^2 = 12.25. Step 4: Then V_cone = (1 / 3) * pi * 12.25 * 12 = (1 / 3) * pi * 147 = 49 * pi. Step 5: Write cylinder volume: V_cylinder = pi * r_y^2 * h_y = pi * 7^2 * h_y = pi * 49 * h_y. Step 6: Equate volumes: 49 * pi = 49 * pi * h_y. Step 7: Cancel 49 * pi from both sides: 1 = h_y. Step 8: Therefore, height of water in the cylinder h_y = 1 cm.

Verification / Alternative check:
Observe that the base area of the cylinder is pi * 7^2 = 49 * pi. Since the cone volume came out exactly 49 * pi cubic centimetres, dividing the cone volume by this base area must give a height of 1 cm. This quick reasoning matches the detailed calculation and confirms the result.

Why Other Options Are Wrong:
0.25 cm and 0.5 cm: These are too small and would give a cylinder volume much smaller than the cone volume.
2 cm and 3 cm: These heights would produce volumes 2 and 3 times larger respectively than the cone volume, which contradicts conservation of volume.

Common Pitfalls:
Common mistakes include mixing up the formulas for cone and cylinder volumes, forgetting the factor 1 / 3 in the cone formula, or failing to square the radius. Some students also try to plug in a numerical value of pi unnecessarily when it cancels out. Focusing on algebraic cancellation and carefully writing each step prevents such errors.

Final Answer:
The height of water in the cylindrical vessel is 1 cm.