Rates of change — Tank with rate constant $m$

Question

Hemispherical bowl radius , volume , and while water flows in and out.

(i) Derive .

(ii) Hence show the fill time from empty to full is .

(iii) Inflow then stops; draining alone satisfies . Show the empty time is .

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) , so .
Simplify the previous line carefully, keeping exact values where possible.(ii) .
Finish the calculation, then check that the result meets the question’s conditions.(iii) .
State the final answer clearly in the form the question requested.Integrate : .

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

Verification: Hence links: (i) DE (ii) fill time (iii) empty time ratio .

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