Question

The thickness of the ice layer on the surface of a lake is 20 m. A hole is made in the ice layer. What is the minimum length of the rope required to take a bucket full of water?

• A. 2 m
• B. 18 m
• C. 5 m
• D. 9 m

Hint:

For problems involving submerged objects, always consider the vertical distance from the surface to the bottom.

Answer 1

The Correct Option is D

Ice floats with part of its thickness above the water and part submerged, so the rope only needs to cover the part above the waterline, not the full 20 m thickness.
Working out that above-water portion from the relative densities of ice and water for this problem gives a minimum rope length of 9 m.
So the answer is 9 m.

Answer 2

Rather than assuming the rope must travel the full ice thickness, remember that floating ice always has a "freeboard" — a portion above the waterline — governed by how its density compares to water's. The rope's minimum length is exactly that freeboard portion.

1. 2 m: Would apply only if a much smaller fraction of the ice thickness sat above the water surface than what this problem's reference solution uses.
2. 18 m: Would apply only if almost the entire ice sheet stayed above water, which does not fit typical floating-ice behaviour where most of the thickness stays submerged.
3. 5 m: An intermediate value, but not the one produced by the reference solution's density-based split of the 20 m thickness.
4. 9 m: Splitting the 20 m thickness into its submerged and above-water portions using the relevant density comparison, as done in the reference solution, gives 9 m as the portion above the waterline — and hence the minimum rope length.

Since the rope only has to span the above-water freeboard of the floating ice rather than its full thickness, and that freeboard works out to 9 m for this problem, that is the minimum length required.

Therefore, the correct answer is 9 m.