A solid right circular cylinder has radius 4 centimetres and height 8 centimetres. It is melted and recast into a right circular cone of the same radius. What will be the height of the cone, in centimetres, assuming no loss of material?

Introduction / Context:
This question involves solid geometry and volume conservation. A cylinder is melted and recast into a cone with the same radius. Such problems appear frequently in aptitude exams to test whether you can apply volume formulas and the idea that volume is conserved when a solid is melted and reshaped without loss of material.

Given Data / Assumptions:

• Radius of the original cylinder r = 4 centimetres.
• Height of the cylinder h1 = 8 centimetres.
• The cylinder is melted and recast into a cone of radius r = 4 centimetres.
• Height of the cone h2 is unknown and must be found.
• No material is lost during melting and recasting, so volumes are equal.
• pi cancels out in the calculations, so we do not need a numeric value.

Concept / Approach:
The volume of a right circular cylinder is Vcyl = pi * r^2 * h1. The volume of a right circular cone is Vcone = (1/3) * pi * r^2 * h2. Because the cylinder is melted and recast into the cone without loss of material, we set these volumes equal: pi * r^2 * h1 = (1/3) * pi * r^2 * h2. The common factors pi and r^2 cancel out, leaving a simple relationship between the heights. We then solve for h2.

Step-by-Step Solution:
Step 1: Write volume of cylinder: Vcyl = pi * r^2 * h1. With r = 4 and h1 = 8, Vcyl = pi * 4^2 * 8 = pi * 16 * 8 = 128 * pi. Step 2: Write volume of cone: Vcone = (1/3) * pi * r^2 * h2. With r = 4, Vcone = (1/3) * pi * 16 * h2 = (16/3) * pi * h2. Step 3: Set volumes equal: 128 * pi = (16/3) * pi * h2. Cancel pi from both sides to get 128 = (16/3) * h2. Step 4: Multiply both sides by 3: 128 * 3 = 16 * h2. So 384 = 16 * h2. Step 5: Divide both sides by 16: h2 = 384 / 16 = 24 centimetres.

Verification / Alternative check:
We can verify by plugging h2 = 24 back into the cone volume formula. Vcone = (1/3) * pi * 4^2 * 24 = (1/3) * pi * 16 * 24 = (1/3) * pi * 384 = 128 * pi. This matches the original cylinder volume Vcyl = 128 * pi. Since the volumes are equal, our computed height of 24 centimetres is correct.

Why Other Options Are Wrong:
Option A (48 cm): This would double the correct height and give cone volume (1/3) * pi * 16 * 48 = 256 * pi, which is too large. Option C (36 cm): This leads to a cone volume larger than 128 * pi and violates volume conservation. Option D (12 cm): This gives cone volume (1/3) * pi * 16 * 12 = 64 * pi, which is only half of the required volume. Option E (18 cm): This also produces a volume different from 128 * pi, so it cannot be correct.

Common Pitfalls:
A common mistake is forgetting to use the factor 1/3 in the cone volume formula or mixing up cylinder and cone formulas. Another issue is trying to plug in pi numerically when it is not necessary, which can introduce rounding errors. Some students also try to conserve surface area instead of volume, which is incorrect for melting and recasting problems. Always remember that when a solid is melted and reshaped, it is the volume that remains constant.

Final Answer:
The height of the cone is 24 cm (24 centimetres).