Answer to the September 6, 2004 Problem of the Week

The Coconut Cake Problem

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One day, my son-in-law Eric brought home a coconut cake and announced that he wanted a piece of cake with the most icing (meaning a corner). The cake was a square cake 6"x6" and 3" tall with icing on the four sides and the top. Consider the thickness of the icing to be negligible.

We decided to cut the cake in four pieces of equal volume so that my son-in-law could have the most icing. We made six cuts shown in red in the diagram below:

Determine how far in from the side of the cake that the cuts should be made. How many more square inches of icing will Eric get than each of the rest of us? (Yes, Eric really cut the cake this way!).

 

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Solution to the Problem:

The cuts should be made .4019 inches from the side of the cake. Eric got 72 square inches more of icing than each of the rest of us.

The first part of this problem can be found by changing this to an area problem and considering just the 6"x6" square (since the depth is always three inches).
The total area of the top square is 36 square inches.
Each person should get 9 square inches of cake on the top.

See the diagram below.

My son-in-law's piece of cake had 81 square inches of icing (9 sq. in. + 4 x 6" x 3");
each of the rest of us had 9 square inches of icing.