Yummy Jelly
A jelly shop sells its products in two different sets: 3 red jelly cubes and 3 green jelly cuboids.
The 3 red cubes are of side lengths a < b < c , while the 3 green cuboids are identical with dimensions a × b × c , as shown above.
Which option would give you more jelly?
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2 solutions
Moderator note:
Simple standard approach.
Interesting approach, Dr. Warm. I solved it in a slightly different way. We know that a 3 + b 3 + c 3 − 3 a b c = 2 1 ( a + b + c ) ( ( a − b ) 2 + ( b − c ) 2 + ( c − a ) 2 )
We see that the RHS is positive since a , b , c are not all equal. Hence the volume of red jellies is (strictly) greater than the volume of the green jellies.
I did it the same way
We know that AM GM inequality could be proved using Cauchy Induction ( or the Forward Backward induction) but can this approach to induction be applied to the other formulae that we can prove with induction ( weak and strong) or it has no other applications.If the former is true I would like to know how?
Sum of the volumes of the three red cubes= a 3 + b 3 + c 3
sum of volumes of the three green cuboids= abc + abc + abc= 3abc
AM >= GM
( a 3 + b 3 + c 3 )/3 >= cube root ( a 3 b 3 c 3 )
a 3 + b 3 + c 3 >= 3abc
a>b>c so equality is not possible because for equality, a=b=c
therefore a 3 + b 3 + c 3 > 3abc
hence, volume of the three cubes > volume of the three cuboids
From the given information, obviously the volume of the red jelly = a 3 + b 3 + c 3 .
On the other hand, the volume of the green jelly = 3 a b c .
According to Arithmetric Mean - Geometric Mean Inequality , 3 a 3 + b 3 + c 3 ≥ 3 a 3 b 3 c 3 = a b c .
Hence, a 3 + b 3 + c 3 ≥ 3 a b c .
Since the inequality will become equation if and only if a = b = c , the volumes of both jelly sets can never be equal.
As a result, the red jelly will have more volume than the green one.