 Hi and how are you all today? The question says an open box with a square base is to be made out of a given quantity of metal sheet of area c square. Show that the maximum volume of the box is c square upon 6 root 3. So here let x be the side of the square. Let y be the height of the box will be 4xy plus x square. We are given c square. So here c square is equal to 4xy plus x square which implies the value of y is equal to c to the volume. This implies on substituting the value of y from here we have in place of y c square minus x square upon which gives us c square minus x square. Now differentiating this equation with respect to x we have dv by d and again differentiating it with respect to x we have d square v by dx square equal to minus x minus x by 2 which gives us minus 3x upon 0. 0 we will obtain the value of x from that and then the second derivative is less than 0. So for that this implies minus x square upon 2 plus this value I have taken out from here is equal to 0. This further implies c square minus x square upon 4 is equal to x square upon 2 simplifying it we have c square minus x square minus 2x square equal to 0. This gives us c square minus 3x square equal to 0 we have c square equal to 3x square this gives us square upon 3 that 2 in the root gives us the value of x which implies the value of x is plus minus c upon root 3. Now the value of the second derivative c upon under root 2 minus 3 c upon which is c upon root 3 point of maxima and so that maximum volume of the box is given by now we will obtain the value of v when x is equal to c upon root 3 and this gives us 1 upon 4 into c upon root 3 bracket c square minus c square upon 3 this gives us the value of the volume that is a cube upon and this is the required answer to the given question. So hope you understood the whole concept well we were need we were required volume of the box is cx root 3 and this is