 Hello and welcome to the session. In this session we discussed the following questions that says, show that the right circular cylinder opened at the top and the given surface area and maximum volume is such that its height is equal to the radius of the base. Let's proceed with the solution now. We have given a right circular cylinder which is opened at the top and we have given the surface area and its volume is maximum such that its height is equal to the radius of the base. This is what we have to show. We suppose that Rb, the radius of the base, the open circular cylinder, Hb, the height of the cylinder, Hb, the surface area, Vb, the volume of the cylinder. So, S is equal to the area of the cylinder which is Pi R square plus H is equal to S minus Pi R square, this will equal the volume of the cylinder. So, V is given as S minus Pi R square plus 2 Pi R. So, we have V is equal to Pi R square from the result 1. It cancels with Pi and cancels with R. V is equal to S minus this whole into R upon 2. Half into SR minus Pi R cube the whole. Since we have to show that the volume is maximum when height is equal to the radius, we differentiate the volume height to the radius. So, differentiating is equal to half into F the whole. Now, really now we take dV by dr equal to 0 that is half into the whole equal to 0 which gives us S minus 3 Pi R square equal to 0. Now, R square equal to 0 which gives us minus 2 Pi R square plus equal to 0 or you can say equal to R square that is 2 Pi cancels with 2 Pi which is equal to R. This means that the height of the cylinder is equal to the radius of the cylinder which is equal to R by dr equal to half into S minus 3 Pi R square the whole by dr equal to half into S minus R square the whole. And from here we have d2v R2 is equal to half into 0 is equal to upon 2 that is 2 is equal to minus d2v by h equal to R plus 3 Pi R which is less than 0. So, there so we can say that the cylinder the height of the cylinder is equal to the radius of the base of the cylinder. And this is what we were supposed to prove. So, here is proved the session that we have understood the solution of this question.