Given that the sum of the lengths of all the edges of a rectangular parallelepiped with a square base is cm, find its largest possible volume.

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Example

Given that the sum of the lengths of all the edges of a rectangular parallelepiped with a square base is cm, find its largest possible volume.

Solution

Let be the side length of one of the square bases of the rectangular parallelepiped, and let be the rectangular parallelepiped's height.
Since the rectangular parallelepiped has 8 sides of length and 4 sides of length , it must be true that . Solve for in terms of as follows:

Now let be the volume of the rectangular parallelepiped.
Since the volume is equal to the area of the rectangular parallelepiped's base times the rectangular parallelepiped's height, it must be true that .
Next, find .

Since is defined for all values of , the critical points of occur when .
Now substitute 0 for in the equation and solve for .

Next, find as follows:

Now substitute 0 into the equation for to find .

Since , it must be true that has a minimum at . Next, substitute 3 into the equation for to find .

Since , it must be true that has a maximum at . Finally, substitute 3 for in the equation to find .

Thus, the largest possible volume of the rectangular parallelepiped is .