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2021 HSC Mathematics Extension 1

Question 12(b):
Differential equation

4 marksdifferential equations

A bottle of water, with temperature , is placed on a table in a room. The temperature of the room remains constant at . After minutes, the temperature of the water, in degrees Celsius, is .

The temperature of the water can be modelled using (do NOT prove this), where is a constant.

(i) After minutes, the temperature of the water is . By solving the differential equation, find the value of when the temperature of the water reaches . Give your answer to the nearest minute.

(ii) Sketch the graph of as a function of .

How to recognise this question

Question type: differential equations

  • The command and notation point to differential equations.
  • The word “hence” means the later part is intended to reuse the earlier result.

How to handle it: separate or integrate the differential equation, include the constant, then use the initial condition.

Watch out: Do not select a formula until its domain, interval, sign and units match the question.

Step-by-step answer

  1. (i) Separate: .
  2. Since starts below ambient and increases toward , . At , : . So .
  3. At , : .
  4. Set : .
  5. Thus minutes.
  6. NESA (i): 3 marks full; 2 for finding (or equivalent); 1 for the general solution or for finding .
  7. (ii) Sketch: , increasing, concave down (warming slows as approaches ), horizontal asymptote as . Passes through and about .
  8. NESA (ii): 1 mark for a correct sketch.

Common mistake: choosing a familiar formula before checking that its conditions, signs, interval and units match this question.

The paper records 4 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

Coffee at is placed in a room that remains at . After minutes its temperature is , modelled by (do NOT prove this).

(i) After minutes, . By solving the differential equation, find when . Give your answer to the nearest minute.

(ii) Sketch the graph of as a function of .

See the step-by-step answer guide →

Question 2

Water at is placed in a room that remains at . After minutes its temperature is , modelled by (do NOT prove this).

(i) After minutes, . By solving the differential equation, find when . Give your answer to the nearest minute.

(ii) Sketch the graph of as a function of .

See the step-by-step answer guide →

Question 3

Tea at cools in a room that remains at . After minutes its temperature is , modelled by (do NOT prove this).

(i) After minutes, . By solving the differential equation, find when . Give your answer to the nearest minute.

(ii) Sketch the graph of as a function of .

See the step-by-step answer guide →