Rates of change — Same tank with constant $\alpha$

Question

A hemispherical tank of radius has volume when the water depth is . While filling and draining, for a positive constant .

(i) Show that .

(ii) Hence show that the tank is full after seconds.

(iii) When full, inflow stops and draining continues with . Show that the tank takes seconds to empty.

This practice question was inspired by Question 13(a) in the 2023 NSW HSC Mathematics Extension 1 examination, © NESA. It is an original TestMum variation that practises the same mathematical idea, not an official NESA question.

Set up the solution: write the geometric constraint, differentiate every changing quantity with respect to time, then substitute.(i) . Chain rule: . Cancel : .
Simplify the previous line carefully, keeping exact values where possible.(ii) . Integrate , : .
Simplify the previous line carefully, keeping exact values where possible.(iii) Drain: .
Finish the calculation, then check that the result meets the question’s conditions.Separate: .
State the final answer clearly in the form the question requested.LHS , so .

Common mistake: changing so many features that the new example no longer tests the same mathematical idea.

Verification: Full three-part fill–empty chain with renamed constant .

Any mark labels are a TestMum study aid, not an official NESA marking allocation.