Water is running in a hemispherical bowl of radius 180 cm at the rate of 108 cubic decimeters per minute. How fast the water level is rising when depth of the water level in the bowl is 120 cm ? ( 1 decimeter =10 ~cm )

Step-by-step solution

Radius of hemispherical bowl ( r )=180 ~cm Rate of flow aligned ( dV dt ) & =108 dm ^3 / min & =108000 ~cm ^3 / min & = 108000 60 ~cm ^3 / sec & =1800 ~cm ^3 / sec aligned Let depth of water in bowl be x . Volume of water in hemispherical aligned & bowl ( V )= 3 x^2(3 r-x) & V = 3 x^2(3 180-x) & V =180 x^2- 3 x^3 aligned Differentiating w.r.t. x , we get aligned & dV dt =360 x ~d x dt - x^2 ~d x dt & . dV dt |_ x=120 = d x dt (360 120-120^2 ) aligned array ll & 1800= d x d t (360 120-120^2 ) & 15= d x d t (360 -120