# 12.8.2. Maximum and Minimum Values

A box with an open top is needed to be constructed by cutting equal squares from the corners of a squared thin metallic lamina whose side length is cm and turning up the sides. Find the length of the side of the removed square if the volume of the box is to be maximum.

• Acm
• Bcm
• Ccm
• Dcm

### Example

A box with an open top is needed to be constructed by cutting equal squares from the corners of a squared thin metallic lamina whose side length is cm and turning up the sides. Find the length of the side of the removed square if the volume of the box is to be maximum.

### Solution

Denote the side of the removed squares as . Then the length and width of the base of the open box will each be as will be removed from each side.

The height of the open box will be .

The volume of the box, , is the product of its length, width and height, and therefore

To find the value of that maximises the volume, first differentiate with respect to .

Stationary points occur when , so set the derivative equal to 0 and solve the resulting quadratic equation for :

The value of is the value that would give the minimum volume, as this would gives us

Therefore, the maximum volume is achieved when the side of the square removed is .

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