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 c3/ 6√3cubic 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
Applying the sum rule of 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
Differentiating equation (iii) again with respect to x, we get
∴ Maximum volume of the box is
Hence the maximum volume of the box is c3/ 6√3 cubic units.
Hence proved