178. For a particular rectangular solid with integer dimensions, the sum of its length, width and
height is 50 cm. What is the absolute difference between the greatest possible volume and
the least possible volume of the solid?

OK, well the minimum volume is easy. If any of the values are zero, the volume will be zero. If "integer dimensions" is supposed to imply "positive non-zero integers", then your minimum volume is achieved by setting two sides to the minimum value of 1, and the remaining side to 48:

1 x 1 x 48 = 48cm^3

Now, there's a general rule that for any set of numbers with a fixed sum, their product is maximised if they're the same. For example, a(1-a) is max when a = (1-a) = 0.5. So the three integer values that are closest to each other, and sum to 50, are 17, 17 and 16:

17 x 17 x 16 = 4624cm^3

Which is the maximum volume for the solid (given integer dimensions).

If your question requires a proof of this, it likely involves some partial differentiation - if that's the case you can re-post under calculus specifying that - but I expect you're just required to find the logical answer.

Mar 21st, 2015

(so of course the absolute difference is 4624 - 0 or 48 depending on whether 0 is allowed as a dimension; strictly speaking the question allows it since 0 is an integer, but I think it's more likely that they intended to mean positive integer and just worded it badly. So in that case the difference is 4576).

Mar 21st, 2015

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