Puzzle #15 – Puddle Puzzle

Imagine an infinite hollow cube balancing on one of its edges. An incompressible liquid is allowed to accumulate in the trough formed by the bottom of the cube.

The cube is allowed to slowly tip, such that one of its faces subtends an angle \theta with the plane on which the cube rests; a cross-sectional view is shown in the diagram below.

tippy tippy cube

The initial height of the liquid’s surface above the plane, when \theta=\frac{\pi}{4}, is s_0.

Question: What is the height of the liquid’s surface above the plane s as a function of \theta?

Hint: if the liquid is incompressible, what can you say about its cross-sectional area?

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