If the product of height and square of the radius of the greatest cone obtained by rotating a right-angled triangle of hypotenuse 2 meters about a side is k 3 3 , then k is equal to

Step-by-step solution

Let, the cone is obtained by rotating about B C From the diagram, the radius of the cone is A B and the height is B C The volume generated, V = 1 3 π r 2 h = 1 3 π 4 - x 2 x For maximum volume, d V d x = 0 ⇒ 4 - 3 x 2 = 0 ⇒ x = 2 3 ∴ V m a x = π 3 4 - 4 3 2 3 = 16 π 9 3 Also, d 2 V d x 2 < 0 Hence, x 4 - x 2 = 2 3 4 - 4 3 = 16 3 3