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

Given: an open box with square base is made out of a cardboard of c² area

To show: the maximum volume of the box is cubic units.

Explanation:

Let the side of the square be x cm and

Let the height the box be y cm.

Then area of the card board used is

A = area of square base + 4× area of rectangle

⇒ A = x²+4xy

But it is given this is equal to c², hence

c² = x²+4xy

⇒ 4xy = c²-x²

Then as per the given criteria the volume of the box with square base will be,

V = base×height

Here base is square, so volume becomes

V = x²y……(ii)

Now substituting equation (i) in equation (ii), we get

Now finding the first derivative of the volume, we get

Taking out the constant terms, we get

Applying the sum rule of differentiation, we get

Taking out the constant terms, we get

Applying the differentiation, we get

Now we will apply second derivative test to find out the maximum value of x, so for that let V’ = 0, so equating above equation with 0, we get

⇒ c²-3x² = 0

⇒ 3x² = c²

Differentiating equation (iii) again with respect to x, we get

Taking out the constant terms, we get

Applying differentiation rule of sum, we get

At , the above equation becomes,

Thus the volume (V) is maximum at

∴ Maximum volume of the box is

Hence the maximum volume of the box is cubic units.

Hence proved