An open box with a square base is to be made out of a given quantity of card board of area c2 square units. Show that the maximum volume of the box is c3/ 6 3 cubic units.

Let x be the side of the square base and y be the height of the open box.
Given, area of metal sheet = C2

Area of open box = Area of base + area of 4 sides

= x2 + 4xy

Now, area of open box is also equal to the area of the metal sheet.

Therefore, C2 = x2 + 4xy

Now, volume of the box = x2y

 

On differentiating (1) w.r.t x, we get

Again, differentiating (2) w.r.t. x, we get

Now, for volume to maximum or minimum, 

 

⇒ C2 =  3x2

Or 

Now, x is the length of the box, so it can't be negative.

Therefore,


Now, at 

Therefore,  V is maximum.

Hence, maximum value of V is 

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