# 12.8.2. Maximum and Minimum Values

The sum of the sides of a rectangular parallelepiped having a square base is cm. Find the dimensions that maximize the volume.

• Acm, cm, cm.
• Bcm, cm, cm.
• Ccm, cm, cm.
• Dcm, cm, cm.

### Example

The sum of the sides of a rectangular parallelepiped having a square base is cm. Find the dimensions that maximize the 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 .

Now substitute 0 into the equation for to find .

Since , it must be true that has a minimum at .

Next, substitute 1 into the equation for to find .

Since , it must be true that has a maximum at .

Finally, substitute 1 into the equation and solve for .

Thus, the dimensions that maximise the volume of the rectangular parallelepiped are by by .

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