A cylindrical container, of height 4 cm and radius of cross section 6 cm is completely filled with water. A solid glass prism with an equilateral triangular cross section is then immersed into the cylindrical container, owing to which, some of the water spilled out. What is the maximum volume, in cubic cm, of the water that could spill out?

The maximum volume of water that spills out equals the maximum volume of the immersed equilateral triangular prism, which is its base area times its height. To maximize this volume within the cylinder of radius 6 cm, the circumradius of the equilateral triangle base must be 6 cm, meaning its side length is 6 multiplied by the square root of 3. The area of this equilateral triangle is (root(3)/4) multiplied by the side squared, which equals 27 multiplied by the square root of 3, so multiplying by the height of 4 cm gives a volume of 108 multiplied by the square root of 3.

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