A hemispherical tank with a radius of 10m is filled from an input pipe at a rate of 3m^3/min. How fast is the water level rising when the water level is 5m from the bottom of the tank? (hint: the volume of the cap of thickness of h sliced from a sphere of radius r is pih^2(3r-h)/3)

Answer 1

v = π/3 h^2 (3r-h) = πrh^2 - π/3 h^3
dv/dt = 2πrh dh/dt - πh^2 dh/dt
= πh(2r-h) dh/dt
Pluggin in your numbers, then, we have

π*5(20-5) dh/dt = 3
dh/dt = 3/(75π) = 0.021 m/min