Rates of change — Parallel hemispherical tank chain

Question

A hemispherical water tank of radius cm has cm at depth cm. During simultaneous fill and drain, .

(i) Show that .

(ii) Hence show the tank fills in seconds.

(iii) At the instant the tank is full, inflow stops and thereafter. Show that emptying takes three times as long as filling.

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) Differentiate: . Then , so .
Finish the calculation, then check that the result meets the question’s conditions.(ii) Separate and integrate from empty to full: .
State the final answer clearly in the form the question requested.(iii) Emptying: . Definite integration yields .

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

Verification: Complete (i)–(iii) structure matching the exam's dependent chain.

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