2023 HSC Mathematics Extension 1
Question 13(a):
Rates of change
A hemispherical water tank has radius cm. Initially the tank is empty. Water is poured in at cm s and also drains. After seconds the water height is cm and volume cm (Do NOT prove this).
While filling and draining, .
(i) Show that .
(ii) Show that the tank is full after seconds.
(iii) The instant the tank is full, inflow stops but draining continues as before. Show that the tank takes times as long to empty as it did to fill.
How to recognise this question
Question type: rates motion
- The command and notation point to rates motion.
- The word “hence” means the later part is intended to reuse the earlier result.
How to handle it: write the geometric constraint, differentiate every changing quantity with respect to time, then substitute.
Watch out: Do not restart the second part with unrelated numbers or discard the result proved in the first part.
Step-by-step answer
- (i) Differentiate with respect to : .
- Chain rule: .
- Given . For , , cancel: .
- NESA (i): 2 marks; 1 for .
- (ii) Separate: . Integrate , : .
- NESA (ii): 2 marks; 1 for separating variables.
- (iii) When full, inflow stops. Drain continues so that (NESA sample).
- Then .
- Cancel : .
- Separate and integrate from full to empty: , .
- .
- LHS .
- RHS , so .
- NESA (iii): 3 marks; 2 for correct ; 1 for modelling .
Common mistake: choosing a familiar formula before checking that its conditions, signs, interval and units match this question.
The paper records 7 total marks for this group. Where the official marking guide provides useful partial-credit criteria, those criteria are stated in the steps above.
Try your hand at it again by answering these questions
Question 1
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.
Question 2
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 .
Question 3
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.