How fast is the water level rising when the water level is 5m from the bottom of the tank?

Question

A hemispherical water tank with a radius of 10 m is filled from an inflow pipe at a rate of 3 m^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 a cap of thickness h sliced from a sphere of radius r is πh^2(3r-h)/3.

Answer 1

With a related rates question like this, I like to identify the pieces that are changing along with how they are related to each other.

1. How is the volume changing over time?
2. Can you write the volume of a cap of thickness h as a function of h only? (No r allowed when you are done.)
3. Can you use the chain rule and implicit differentiation to solve for h'(t)?
4. Finally, what is h'(t) when h is 5m ?