A solid S is made from a cylinder surmounted by a hemisphere on top with both its circular faces sharing a common centre. The radius of cylinder and radius of hemisphere are x ~cm . The height of the cylinder is (20-4 x) cm and the volume of S is V= 1 3 y . Find the maximum value of y .

Step-by-step solution

The volume V of the solid S is the sum of the volume of the cylinder and the volume of the hemisphere. Volume of cylinder = r^2 h = x^2 (20 - 4x) Volume of hemisphere = 2 3 r^3 = 2 3 x^3 Total volume V = x^2 (20 - 4x) + 2 3 x^3 = (20x^2 - 4x^3 + 2 3 x^3) = (20x^2 - 10 3 x^3) Given V = 1 3 y , we have 1 3 y = (20x^2 - 10 3 x^3) , which implies y = 60x^2 - 10x^3 . To find the maximum value of y , differentiate with respect to x : dy dx = 120x - 30x^2 . Setting dy dx = 0 , we get 30x(4 - x) = 0 . Since x > 0 , we have