SOLUTION: Volumes of Cubes & Dimensions? The combined volumes of the two cubes with integer side lengths are numerically equal to the combined lengths of their edges. What are the dimensions

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Question 559867: Volumes of Cubes & Dimensions? The combined volumes of the two cubes with integer side lengths are numerically equal to the combined lengths of their edges. What are the dimensions of the cubes?
Answer by richard1234(7193) About Me  (Show Source):
You can put this solution on YOUR website!
Let a and b be the side lengths of the cubes. Since a cube has 12 edges, we have



The LHS and RHS can both be factored:





We can solve this as a quadratic in terms of a:





We want -3b^2 + 48 to be a perfect square (then we must also check that it satisfies the constraints for a to be an integer, but we'll do that later). Fortunately, we do not have to check many values of b since -3b^2 + 48 becomes negative when b >= 5. It can be checked that b = 2 and b = 4 are the only values that yield perfect squares. The corresponding values for a are a = 4 and a = 2 respectively. These two are equivalent (4,2 and 2,4) so the dimensions of the cubes are 4 and 2.