In an invetred cone of 5cm radius and 12cm height a sphere is placed.If the highest point of the sphere touches the base of the cone then find the radius of the sphere.Show that the ratio of volumes of sphere and the cone is 40:81

Volume of cone = (1/3)Pi x r^2h = (1/3)Pi x 5^2 x 12 = 100 Pi Volume of sphere = (4/3)Pi x 6^3=128 Pi You can find the ratio of volumes = 128/100=32/25

Let the radius of the sphere = r cm.OD = OE = rHeight of the cone EA = 12 cmSlant height AC = √{ 52 + 122 } = √169 = 13 cm.In ΔAEC and ODA we have∠AEC = ∠ODA = 90°⇒ ΔAEC ≃ ODA [ AA similarity ] ⇒ OD / OA = EC / AC⇒ r / ( 12 - r) = 5 / 13⇒ r = 10 / 3 cm∴ Volume of the sphere / Volume of the cone = ( 4/3)πr3 / (1/3)πr2h= (4 x 10/3 x 10/3 x 10/3) / (5x5x12)= 40 / 81Volume of the sphere : Volume of the cone= 40 : 81