SOLUTION: A rectangular solid with a square base has a surface area of 37.5 square centimeters. ( let x represent the length of the sides of the square base and let y represent the height.)

Question 1068080: A rectangular solid with a square base has a surface area of 37.5 square centimeters. ( let x represent the length of the sides of the square base and let y represent the height.)
(a) Determine the dimensions that yield the maximum volume.
x=
y=
(b) Find the maximum volume.
Answer by KMST(5328) (Show Source): You can put this solution on YOUR website!
THE QUICK ANSWER:
The maximum area within a given perimeter is the most symmetrical shape you are allowed to choose.
The maximum volume within a given surface area is the most symmetrical shape you can choose.
In this case, the most symmetrical shape you can choose with planar perpendicular faces is a cube.
A cube has congruent faces,
so in a cube with a total surface area of ,
each square face has a surface area of
.
The length of the edge of such a square face is
.

THE EXPECTED SOLUTION:
With and in cm, the surface area, in square cm, is
<---> ,
and the volume, in cubic cm, is

That derivative will be zero only for , and ,
and changes sign from positive to negative only at ,
meaning that is maximum for .
Substituting for in
we find

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