A solid right circular cone of radius 3.5 cm and height 9.6 cm is completely melted and recast into a right circular cylinder of the same radius 3.5 cm. Taking π = 22/7, what will be the height of the resulting cylinder?

Introduction / Context:
This is a volume conservation problem in mensuration involving a cone and a cylinder. A solid cone is melted and its material is used to form a cylinder that has the same radius as the original cone. Since there is no gain or loss of material, the volume of the cone must equal the volume of the cylinder. Using standard volume formulas for a cone and a cylinder, we can find the unknown height of the cylinder.

Given Data / Assumptions:
- Radius of the original cone r = 3.5 cm. - Height of the cone h_cone = 9.6 cm. - The cone is melted and recast into a cylinder with the same radius r = 3.5 cm. - Height of the cylinder is H (unknown). - Volume of a cone: V_cone = (1 / 3) * π * r^2 * h_cone. - Volume of a cylinder: V_cyl = π * r^2 * H. - π is taken as 22 / 7.

Concept / Approach:
The guiding principle is conservation of volume: volume of cone equals volume of cylinder. Because the radius is the same for both solids, the factor π * r^2 appears in both volume formulas and cancels when equated. This simplifies the relation to a direct proportionality between the heights of the two shapes, allowing us to solve for H using a simple division without complicated numerical computation.

Step-by-Step Solution:
Step 1: Write the volume of the cone as V_cone = (1 / 3) * π * r^2 * h_cone. Step 2: Write the volume of the cylinder as V_cyl = π * r^2 * H. Step 3: Since the cone is melted and recast into the cylinder, set V_cone = V_cyl. Step 4: So (1 / 3) * π * r^2 * h_cone = π * r^2 * H. Step 5: Cancel π * r^2 from both sides. This gives (1 / 3) * h_cone = H. Step 6: Substitute h_cone = 9.6 cm, so H = (1 / 3) * 9.6. Step 7: Compute H = 9.6 / 3 = 3.2 cm. Step 8: Therefore, the height of the cylinder is 3.2 cm.

Verification / Alternative check:
We can estimate volumes numerically to check. Volume of the cone is approximately V_cone = (1 / 3) * π * 3.5^2 * 9.6. The cylinder volume with H = 3.2 is V_cyl = π * 3.5^2 * 3.2. Comparing, note that V_cyl is exactly one third of π * 3.5^2 * 9.6, which matches V_cone. This confirms that H = 3.2 cm satisfies the volume equality exactly, so no arithmetic error has occurred.

Why Other Options Are Wrong:
Option 6.4 cm would double the height of the cylinder and hence double its volume compared to what the cone provides. Option 1.6 cm would produce a cylinder with only half the correct volume, meaning not all cone material is used. Option 4.8 cm would give a cylinder volume equal to 1.5 times the cone volume, which is not possible under conservation of material.

Common Pitfalls:
Students sometimes mistakenly equate only the heights or only the radii without using the full volume formulas. Another common error is to forget the 1 / 3 factor in the cone volume formula, which completely changes the result. Also, some might try to carry π through unnecessary calculations instead of cancelling it early. Recognising that identical radii greatly simplify the ratio of volumes is an important problem solving skill in mensuration.

Final Answer:
The height of the cylinder formed from the melted cone is 3.2 cm.