Cheating With A Valentine Cake

Geometry Level 5

A mother brings home a surplus Valentine Cake from the bakery where she works.

It's made from 3 semi-circles of radii 2, 3, 5 inches, 3 inches thick. See graphic

It has a rich red frosting on top and on the vertical sides. She decides to feed it to her two boys. But she knows that they are going to both want the same amount of cake and the same amount of frosting. She realizes that she cannot divide the cake in half with one straight vertical cut into just 2 pieces that both have the same cake volume and frosting surface area. So, picking a point on the perimeter of the cake, she makes 2 straight almost vertical cuts from it (the 2 cuts meet at the point she picks) dividing the cake into 3 pieces, and takes out the middle piece for herself, leaving 2 pieces for the boys that have exactly the same cake volume and frosting surface area. Let \(V<10\) be the smallest cake volume of the piece she can cut out for herself. Cake is 3 inches thick.

Find \(\left\lfloor 100\cdot V \right\rfloor \).

Note: Any piece that contains the cusp of the cake can be considered to be a single piece, i.e., e.g. "two pieces" with the cusp point in common is one piece. Also, "vertical" refers to the \(z\) axis of this solid.

Unhelpful Hint: You may want to look up the Ham Sandwich Theorem, of which the Pancake Theorem is the 2D version of it. According to these theorems, it should be possible for the mother to divide the cake in half with a single straight vertical cut, each half having the same volume and surface area, but here this doesn't apply. Why?

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