An open-top box is to be made by cutting small congruent squares from the corners of a 9 cm * 9 cm sheet of metal and bending up the sides. what is largest possible volume of such a box

Let the length of the side of the 4 small squares be = x.

The formula for the volume of the box will be

height * width * length

V = x (9 - 2x) ^2

V = 81x - 36x^2 + 4x^3

finding the derivative:-

dV / d x = 12x^2 - 72x + 81

THis equals 0 for a maximum / minimum value

12x^2 - 72x + 81 = 0

3 (4x^2 - 24x + 27) = 0

x = 4.5, 1.5

Use second derivative to find maxm and minm:-

d^2V / dx^2 = 24x - 72

when x = 1.5 this is negative and when x = 4.5 this is positive

so x = 1.5 gives a maximum value for V

V = 1.5 (9 - 2 (1-5)) ^2 = 54

Largest possible volume of the box is 54 cm^3 Answer