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 `(c^3)/(6sqrt(3))` cubic units.

Explanation

As base of the box is square,

So, area of the box `= a^2+4ah`

Here, `a` is the side of square base and `h` is the height.

Now, it is given that,

`c^2 = a^2+4ah`

`=>h =1/4 [(c^2-a^2)/a]`

Now, Volume of the box`(V) = a*a*h = a^2h`

`=>V = 1/4 [(c^2-a^2)/a]a^2`

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