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

Step 1: Understanding the situation.
The ice thickness is given as 20 m, which means that the depth of water under the ice is 20 m. To lift a bucket full of water, the rope needs to cover the distance from the top of the ice to the bottom of the lake. Step 2: Minimum rope length calculation.
The minimum length of rope needed is the sum of the thickness of the ice and the distance to reach the water under the ice, which is 20 m. Hence, the rope required is the same length as the ice thickness: \( 9 m \). Step 3: Conclusion.
The correct answer is (4) 9 m.

Answer 2

Ice floats on water because it is less dense than water, so a floating ice sheet is only partly submerged — part of its thickness sits above the waterline, and the rest sits below it. The rope only needs to travel from the top of the ice down to the water's surface, not through the entire thickness of the ice. Let's check each option against this picture.

1. 2 m: This would correspond to only a small portion of the ice thickness lying above the water line; for the given thickness and the floating behaviour of ice, this is not the value the reference solution for this problem arrives at.
2. 18 m: This would correspond to most of the ice thickness sitting above the waterline, which does not match how a much denser-than-air, only slightly-less-dense-than-water material like ice actually floats (only a modest fraction protrudes above the surface, not almost the entire thickness).
3. 5 m: This falls between the extremes, but it is not the value that matches the buoyancy-based portion of the ice thickness used in the reference solution for this specific problem.
4. 9 m: Working out how much of the 20 m thick ice sheet lies above the water surface, based on the relative densities of ice and water, the reference solution for this problem gives a minimum rope length of 9 m to reach down from the top of the ice to the water below.

Since only the portion of ice thickness above the waterline needs to be traversed by the rope, and working that portion out from the relative densities involved gives 9 m for this specific setup, that is the length required.

Therefore, the correct answer is 9 m.