 Hi and welcome to the session. Let's work out the following question. The question says show that the height of the cone of maximum volume that can be inscribed in a sphere of radius 12 centimeter is 16 centimeter Let us start with the solution to this question First of all, let height of this cone be h centimeter. Let omv is equal to x So we will have h to be equal to 12 plus x. Now, this is a cone that is inscribed in this sphere with radius 12 centimeter now radius of Base of the cone That is am will be equal to square root of oa square minus om square That is equal to square root of oa square is 12 square. So 144 minus x square therefore volume will be equal to 1 by 3 into pi into Square of radius into height So this will be equal to 1 by 3 into pi into square root of 144 minus x square the whole square into h that is 12 plus x now for maxima or minima or maximum or minimum value dv by dx should be equal to 0 therefore 1 by 3 pi into 144 minus x square Plus 12 plus x into minus 2x should be equal to 0 this implies that 2x into 12 plus x should be equal to 144 minus x square This implies 24x plus 2x square minus 144 plus x square should be equal to 0 and this implies 3x square Plus 24x minus 144 should be equal to 0 taking 3 common we get 3 into x square plus 8x minus 48 is equal to 0 This implies x square plus 8x minus 48 is equal to 0 now splitting the middle term We have x square plus 12x minus 4x minus 48 is equal to 0 This implies that x into x plus 12 minus 4 into x plus 12 is equal to 0 this implies x plus 12 into x minus 4 is equal to 0 and This further implies that x is equal to minus 12 or 4 now rejecting the negative value We have x is equal to 4 now From this that is dv by dx is equal to a is this so dv by dx now becomes 1 by 3 into pi into 144 minus x square plus minus 24x minus 2x square Which is equal to pi by 3 into 144 minus x square minus 24x minus 2x square This is equal to pi by 3 into minus 3x square minus 24x plus 144 now we see that d2 v by dx 2 that is the second derivative is equal to pi by 3 into minus 6x minus 24 Now d2 v on dx 2 At the point x equal to 4 is equal to pi by 3 into minus 6 into 4 minus 24 Which is further equal to pi by 3 into minus 24 minus 24 Which is equal to pi by 3 into minus 48 that is equal to minus 16 pi Which is negative? Therefore we can say that maxima occurs at x equal to 4 and therefore height of this cone That is h will be equal to 12 plus x that is 12 plus 4 that is equal to 16 centimeter So this is what we were supposed to prove in this question I hope that you understood the solution and enjoyed the session. Have a good day