A solid right circular cylindrical metal rod has a radius of 3.2 decimetres. It is completely melted and recast into 44 equal solid cubes, each having an edge length of 8 cm. What is the length (in cm) of the original cylindrical rod? (Take π = 22/7.)

Introduction / Context:
This problem tests your understanding of volume conservation when a solid is melted and recast into new shapes. The cylindrical rod and the cubes have the same total volume of material, so the volume of the cylinder must equal the sum of the volumes of all cubes.

Given Data / Assumptions:

• Radius of the cylindrical rod r = 3.2 dm.
• 1 dm = 10 cm, so r = 32 cm.
• The rod is melted into 44 solid cubes.
• Edge of each cube = 8 cm.
• Take π = 22/7.

Concept / Approach:
We use the formula for volume of a cylinder, V = π * r^2 * h, and for a cube, V = a^3. Since the rod is melted and recast, total volume of the cylinder equals the sum of volumes of the 44 cubes. From this equality we solve for the unknown height h (length) of the rod in centimetres.

Step-by-Step Solution:
Step 1: Convert radius from decimetres to centimetres: r = 3.2 dm = 32 cm.Step 2: Volume of one cube = 8^3 = 512 cubic cm.Step 3: Total volume of 44 cubes = 44 * 512 = 22528 cubic cm.Step 4: Let h be the length of the cylindrical rod in cm. Volume of cylinder = π * r^2 * h = (22 / 7) * 32^2 * h.Step 5: Equate volumes: (22 / 7) * 32^2 * h = 22528.Step 6: Compute 32^2 = 1024, so (22 / 7) * 1024 * h = 22528.Step 7: Simplify: (22528 * 7) / (22 * 1024) = h, which evaluates to h = 7 cm.

Verification / Alternative check:
You can quickly check by substituting h = 7 cm back into the cylinder volume formula and confirming that the result is 22528 cubic cm, which matches the combined volume of the 44 cubes. This confirms the calculation is consistent.

Why Other Options Are Wrong:
56 cm and 5.6 cm give cylinder volumes that are far too large or too small compared to 22528 cubic cm. The option 0.7 cm is off by a factor of ten and clearly unrealistic. Only 7 cm exactly matches the required equality of volumes.

Common Pitfalls:
Students often forget to convert decimetres to centimetres or misapply π. Another common mistake is to multiply instead of equating total volumes, or to forget that all 44 cubes together form the cylinder material. Carefully managing units prevents these errors.

Final Answer:
The length of the original cylindrical rod is 7 cm.