 Hi and how are you all today? My name is Priyanka and I shall be helping you with the following question it says Show that a right circular cylinder which is open at the top and has a given surface area Will have the greatest volume if its height is equal to the radius of its base Let's proceed with our solution here. Let us consider a right circular cylinder with open top radius of base as R right for the let's surface area of Cylinder though that will be 2 pi R H that is the curve surface area of the cylinder Plus area of its base since it is having an open top and that is pi R Square taking out pi R common we are left with 2 H plus R Which can be rearranged and written as S minus pi R square divided by 2 pi R is equal to the height Right now we know that Volume of this cylinder is Equal to be that is pi R square H now in place of H to make it in one variable We can write it as V is equal to pi R square now in place of H. We can write down S minus pi R square divided by 2 pi R which is on simplifying We have Volume equal to pi R Bracket S minus pi R square divided by 2 pi now differentiating with respect to R we get TV by DR is equal to D by DR of Pi R S minus pi R square divided by 2 pi now we have It further written as DV by DR is equal to D by DR Now again simplifying we are left with 1 by 2 is the constant We'll take it out of the Brackets and hence taking out of the derivative we have 1 by 2 Bracket now using the product rule here. We have the first function This is the first function and this is the second function. So we have first function into derivative of second That will be minus 2 pi R plus second function Into derivative of first function, which is one. So we have DV by DR equal to 1 by 2 Here we have minus 2 pi R square minus S sorry positive S minus pi R square These two are like them. So we will add them. We have minus 3 pi R square plus S Which can also be written as DV by DR is equal to 1 by 2 S minus 3 pi R square We know that volume is maximum when the first derivative is equal to 0 and the second derivative is Is less than 0. So let us put DV by DR equal to 0 This implies 1 by 2 into S minus 3 pi R square is equal to 0 further implies That we have S equal to 3 pi R square which is the value of R is under root S by 3 pi Now let us find out the second derivative Here it will be D square V by DR square is equal to D by DR of the first derivative which is D by DR of Let us open there. We have S by 2 minus 3 by 2 pi R square So we have D square V by DR square is equal to this is a constant So it will be 0. So we have minus 3 by 2 into 2 pi R into 1 Which is equal to minus 3 pi R, which is less than 0. So this implies that volume of this cylinder is maximum When R is equal to under root S by 3 pi Now here in this question, we need to prove that this have this right circular Cylinder has the greatest volume if its height is equal to the radius of its base Right now here we have R as under root S by 3 pi Let us find out the value of H that is height above We said that H is equal to S minus pi R square divided by 2 pi R. This is what we found out in the first step Here and then we substituted it in the volume, right? Remember? So now let us put back the value of R here. So we have H equal to S minus pi square root Into the whole square will give us S by 3 pi itself divided by 2 pi Under root S by 3 pi Simplify it We have H equal to 3 S minus S by 3 The whole divided by or we can also write it like divided by 2 pi under root S by 3 pi Which is now further equal to H is equal to 2 S by 3 into Under root 3 pi whole divided by 2 pi under root S. So we have this 3 We'll get Multiplied we are left with root 3 over here. This under root pi will get divided we are left with under root pi and here we are left with under root S So we have H equal to under root S by under root 3 into 1 by under root pi which can be clubbed and written as that H is equal to under root S divided by 3 pi which is same as The radius so now we can write our answer that when radius is equal to height is equal to under root S by 3 pi then volume is Maximum right so this completes the session. Hope you understood the whole concept well and enjoyed it to have a nice day ahead