Question

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

Answer 1

The Maximum volume of the box is: c³/6√3 cubic units.

How to find the maximum value of the box?

Let the length, breadth and height of open box with square base be x, x and h unit respectively. If V be the volume of box then:

V = x * x * h

V = x²h ----(1)

Also:

c² = x² + 4xh

h = (c² - x²)/4x

Putting it in eq 1 gives:

V = x² * (c² - x²)/4x

V =
`(c^(2)x )/(4) - (x^(3) )/(4)`

Differentiating with respect to x gives:

dV/dx =
`(c^(2) )/(4) - (3x^(2) )/(4)`

At dV/dx = 0, we have:

x = c/√3

dV²/dx² will give a negative value and as such:

x = c/√3 gives a maximum volume

Putting x = c/√3 into h = (c² - x²)/4x gives:

h = c/2√3

Thus:

Maximum volume = (c/√3)² * c/2√3

Maximum volume = c³/6√3

Complete question is:

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