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: a cardboard box that is open and square in shape has area
To show: cubic units is the maximum volume of the box.
Explanation:
Take the side of the square be x cm and
Take the height the box be y cm.
So, the total area of the cardboard used is
A = area of square base + 4x area of rectangle
But it is given this is equal to , hence
According to the given condition the area of the square base will be
V = base × height
Since the base is square, the volume is
Then putting the values of equation (i) in equation (ii), we get
Calculation of the first derivative of the equation,
Removing all the constant terms
Using the sum rule of differentiation, we get
Removing all the constant terms, we get
After differentiating the equation, we get
We need to calculate the second derivative to find out the maximum value of x , so for that let equating above equation with 0, we get
Differentiating equation (iii) again with respect to x, we get
Removing all the constant terms results into
Using the differentiation rule of sum, we get
, the above equation becomes,
Thus, the volume (V) is maximum at
∴ The box has a maximum value of
So, the box has a maximum value of is cubic units.
Hence, proved.