A right circular cone has height 9 ~cm and radius of base 5 ~cm . It is inverted and water is poured into it. If at any instant, the water level rises at the rate A cm / sec . where A is area of the water surface at that instant, then cone is completely filled in

Step-by-step solution

For the conical vessel, h =9 ~cm , r =5 ~cm Full volume of the vessel, aligned V & = 1 3 r ^2 ~h & = 1 3 25 9 & =75 cm ^3 aligned Now, h r = 9 5 aligned & r = 5 ~h 9 & A = r ^2= 25 ~h ^2 81 aligned According to the given condition, dh dt = A = 81 25 ~h ^2 = 81 25 ~h ^2 h^2 dh = 81 25 dt Integrating on both sides, we get aligned & h ^3 3 = 81 25 t + c ₁ & h ^3= 243 25 t + c , where c =3 c ₁ aligned Naturally, h =0 , when t =0 and hence, c =0 aligned h ^3 & = 243 25 t V & = 1 3 r ^2 ~h & = 1 3 25 ~h ^2 81 ~h & = 25 2