 Hi and welcome to the session. Let us discuss the following question. Question says, a rectangular sheet of tin 45 cm by 24 cm is to be made into a box without top by cutting off square from each corner and folding up the flaps. What should be the side of the square to be cut off so that volume of the box is maximum? First of all let us understand that if we are given a function f. Defined on interval i, c belongs to interval i such that f double dash c exists. Then x is equal to c is a point of local maxima if f dash c is equal to 0 and f double dash c is less than 0. Value fc is local maximum value of function f. This is the key idea to solve the given question. Let us now start with the solution. We are given a rectangle whose length is 45 cm and breadth is 24 cm. Now let us assume that each side of the square which is cut off from the corner be x cm. So we can write let each side of the square cut off from the corner is equal to x cm. Now clearly we can see length of the box is equal to 45 minus 2 x cm, breadth of the box is equal to 24 minus 2 x cm and height of the box is equal to x cm. So we can write length of the box is equal to 45 minus 2 x cm, breadth of the box is equal to 24 minus 2 x cm, height of the box is equal to x cm. Now volume of the box is given by length into breadth into height. Now volume of box v is equal to 45 minus 2 x multiplied by 24 minus 2 x multiplied by x. Now differentiating both sides with respect to x we get dv upon dx is equal to 45 minus 2 x multiplied by 24 minus 2 x multiplied by 1 plus 45 minus 2 x multiplied by x multiplied by minus 2 plus 24 minus 2 x multiplied by x multiplied by minus 2. Here we have applied the product rule to find derivative of this term. Now simplifying we get dv upon dx is equal to 1 0 8 0 minus 90 x minus 48 x plus 4 x square minus 90 x plus 4 x square minus 48 x plus 4 x square. Now this is further equal to 12 x square minus 276 x plus 1 0 8 0. Now clearly we can see minus 90 x minus 48 x minus 90 x minus 48 x is equal to minus 276 x and 4 x square plus 4 x square plus 4 x square is equal to 12 x square. So we get dv upon dx is equal to 12 x square minus 276 x plus 1080. Now we will find all the values of x at which dv upon dx is equal to 0. So we will put dv upon dx is equal to 0. This implies 12 x square minus 276 x plus 1 0 8 0 is equal to 0. Now dividing both sides by 12 we get x square minus 23 x plus 90 is equal to 0. Now this implies x minus 5 multiplied by x minus 18 is equal to 0. Facturizing this quadratic equation we get these two factors x minus 5 and x minus 18. Now this implies x minus 5 is equal to 0 or x minus 18 is equal to 0. Now if we add 5 on both the sides we get x is equal to 5 and here if we add 18 on both the sides we get x is equal to 18. So we get x is equal to 5 or x is equal to 18. Now we know breadth of the box is equal to 24 minus 2x. At x is equal to 18 it will become negative. So we will reject value of x is equal to 18. So neglecting x is equal to 18 we get x is equal to 5. Hence only value of x is 5. Now to show that volume is maximum at x is equal to 5 we will find second derivative of v. We know dv upon dx is equal to 12x square minus 276x plus 1080 this we have already shown above. Now differentiating both the sides with respect to x again we get d square v upon dx square is equal to 24x minus 276. Now we will find the value of d square v upon dx square at critical point that is at x is equal to 5 this is equal to 24 multiplied by 5 minus 276. Now this is further equal to 120 minus 276 which is equal to minus 156. So we get d square v upon dx square at x is equal to 5 is less than 0. Now we know at x is equal to 5 dv upon dx is equal to 0 and d square v upon dx square is less than 0. So this implies x is equal to 5 is a point of local maxima or we can say volume of the box is maximum at x is equal to 5. Now we know side of the square to be cut off is x centimeter and volume is maximum at x is equal to 5. So side of the square to be cut off is equal to 5 centimeter. So our required answer is x is equal to 5 centimeter. This completes the session hope you understood the session take care and have a nice day.