A square cake 6 in by 6 in and 3 in tall was cut into four pieces of equal volume as shown in the figure. Determine how far in (in inches) from the side of the cake the cuts should be made? (i.e., $$x = $$ ?)
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The height is the same everywhere, so equal volumes mean equal areas of the top face, each $$\frac{36}{4} = 9$$ square inches. Piece I is the outer frame of width $$x$$, and the inner square of side $$6 - 2x$$ holds the other three pieces, so $$(6 - 2x)^2 = 27$$. Then $$6 - 2x = 3\sqrt{3}$$ and $$x = 3 - \frac{3\sqrt{3}}{2} \approx 0.40$$ in.