Internet Math Challenge

Solution to the puzzle for 1 November 2002

The height of the yellow part in the newly rearranged candy corn would be cuberoot(19)/3 inches, or about 0.889 inches. (That's nearly 90% of the total height of the cone!)

The easiest way to do this one is to get a formula for the volume of a cone with these proportions in terms of its height h. A cone with height h and base radius r will have volume (1/3)(pi)r2h (where r is the radius of the circular base of the cone). Our cones have radius equal to one-fourth of their height, so we get

V(h) = (1/3)(pi)(h/4)2h = [(pi)/48]*h3.

Now:

Yellow volume = (total volume) - (combined white and orange volume)
= (cone of height 1) - (cone of height 2/3)
= V(1) - V(2/3)
= (pi)/48 - [(pi)/48]*(2/3)3
= [(pi)/48]*(1 - 8/27)
= [(pi)/48]*(19/27).

Since the yellow part of the rearranged candy piece will form the top of the cone, the question becomes "what height of cone gives us this volume?":

V(h) = [(pi)/48]*(19/27)
[(pi)/48}*h3 = [(pi)/48]*(19/27)
h3 = 19/27
h = cuberoot(19/27) = cuberoot(19)/3.


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