 Hi and welcome to the session. Let's work out the following question. The question says a square piece of tin of side 18 centimeter is to be made into a box without top by cutting square piece from each corner and folding up the flaps. What should be the side of square to be cut off so that the volume of the box be maximum also find the maximum volume of box. So let us start with the solution to this question. In the question we are given side of square piece is equal to 18 centimeter let X be the side of the square piece is to be cut off. Let V be the volume of the box so formed then V is equal to 18 minus 2X the whole square into X. This happens because we see that suppose this is the square piece of tin. Now it is of side 18 centimeter now it is to be made into a box without the top by cutting square pieces from each corner. So suppose we cut these square pieces from each corner and let their side be X X and so on. Now since volume is given by length into breadth into height or we can say length will be 18 minus 2X similarly breadth will be 18 minus 2X into height will be X so this is how we get this. Therefore we can say that dV by dx is equal to 18 minus 2X the whole square into 1 plus 2 into 18 minus 2X into minus 2 into X. This is equal to 18 minus 2X the whole square minus 4X into 18 minus 2X this is equal to 18 minus 2X into 18 minus 2X minus 4X this is equal to 18 minus 2X into 18 minus 6X. Now, for maximum or minimum values, dv by dx should be equal to 0. This implies 18 minus 2x should be equal to 0 or 18 minus 6x should be equal to 0. Now this implies that x is equal to 9 cm and this implies that x is equal to 3 cm but x equal to 9 cm is not possible because if we cut 9 cm, 9 cm from each side that will be 18 cm then we would not be able to form a box out of it. So therefore we can say that x is equal to 3 cm. Now the second derivative that is d2v by dx2 will be equal to 18 minus 2x into minus x plus 18 minus 6x into minus 2. Here we apply the product rule. This is equal to minus 144 plus 24x. Now d2v by dx2 at x equals to 3 is equal to minus 144 plus 24 into 3 that is equal to minus 72 which is negative or we can say which is less than 0 therefore volume is maximum x equal to 3 therefore the maximum volume is equal to 18 minus 2 into 3 the whole square multiplied by 3 that is equal to 144 multiplied by 3 that is equal to 432 cm cube. So our answer to this question is that volume is maximum at x equal to 3 and maximum volume is 432 cm cube. So I hope that you understood the solution and enjoyed the session. Have a good day.