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
Now, x is the length of the box, so it can't be negative.
Now, at ,
Therefore, V is maximum.
Hence, maximum value of V is