 Hello and how are you all today? My name is Priyanka and the question says, an open tank with a square base and vertical sides is to be constructed from a metal so as to hold a given quantity of water. Show that the cost of the material will be at least where the depth of the tank is half of its width. Now let's proceed here. First of all, let the side of the square base be equal to x and the height, the open tank be equal to h and the volume of the tank be equal to v. Right, so we can say that volume of the tank is equal to x into x into h which is equal to where h we can write as v upon x squared. Now let the cost per square meter of sheet be rupees p. So the total cost c is given by c is equal to x into x plus x into h plus x into h plus x into h into p. So we have c as x squared plus 4x h into p where we can write now p as p will now can be written as sorry c can now be written as x squared plus 4x into h which is equal to x squared plus 4v upon x. Now differentiating both sides we get with respect to x dC by dx is equal to sorry that p along with it also minus 4v upon x square since p is the constant. Now for maximum or minimum value dC by dx should be equal to 0. So by substituting it equal to 0 we can find out the value of x. It is 2x is equal to 4v upon x square that implies 2x cube is equal to 4v. Further x cube is equal to 2v. Further x is equal to 2v raised to the power 1 by 3. Again differentiating it with respect to x we have d squared c upon dx squared equal to p bracket 2 plus 8v upon x cube. Now in place of x cube we can write 2v so we have p 2 plus 8v upon 2v because x cube is equal to 2v to p 2 plus 4 which is 6p which is greater than 0. So we can write that therefore c will be minimum when x cube is equal to twice the volume and from above we can say that h was v upon x square which is on multiplying both the sides by x we have. x v upon x cube that is x v upon 2v so h is x upon 2. So the cost will be least if depth of the tank is half the width of the base. Right so this completes the solution hope you understood it have a nice day.