Class 12 application of derivatives

Question

A box with a square base is to have an open top . The area of the material for making the box is 192 sq cm . What should be its dimensions in order to the volume of the box is as large as possible..?

Answer 1

Let x = the length of the sides of the box's square base and h = the box's height.  The surface area (SA) of the box is:

 

SA = (area of square base) + (area of 4 sides)

192 = x² + 4xh

(192-x²)/4x = h

 

The volume (V) of the box is:

 

V = x²h

V = x²(192-x²)/4x

V = 48x - x³/4

 

To find the maximum volume, take the derivative of V wrt x, set it to zero, and solve for x.  Once you have x, use the formula for h above to find the height.