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2022 HSC Mathematics Extension 1

Question 12(d):
Differential equation

4 marksdifferential equationsdependency-safe group

In a room with temperature , coffee is poured into a cup. The temperature of the coffee when it is poured is .

The temperature (°C) of the coffee, minutes after it is made, is modelled by , where is a constant of proportionality and is a constant.

(i) It takes 5 minutes for the coffee to cool to . Using separation of variables, solve the differential equation to show that .

(ii) The optimal drinking temperature is . Find when the coffee reaches this temperature, to the nearest minute.

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 restart the second part with unrelated numbers or discard the result proved in the first part.

Step-by-step answer

  1. (i) As , approaches room temperature, so . Also cooling implies in the usual sign convention, but the algebra proceeds with free.
  2. Separate: . Integrate: , so with (here ).
  3. , , so .
  4. , .
  5. Hence for .
  6. NESA (i): 3 marks; 2 for exponential form of ; 1 for separating variables.
  7. (ii) Set : .
  8. Take ln: minutes.
  9. NESA (ii): 1 mark for the correct nearest-minute answer.

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

In a room at , coffee is poured at . The temperature (°C) after minutes satisfies .

(i) After 10 minutes the coffee has cooled to . Using separation of variables, solve the differential equation to show that .

(ii) Hence find when the coffee reaches , to the nearest minute.

See the step-by-step answer guide →

Question 2

In a room at , coffee is poured at . Model by .

(i) After 4 minutes the coffee has cooled to . Using separation of variables, show that .

(ii) Hence find when the coffee reaches , to the nearest minute.

See the step-by-step answer guide →

Question 3

In a room at , coffee is poured at . Model by .

(i) After 6 minutes the coffee has cooled to . Using separation of variables, show that .

(ii) Hence find when the coffee reaches , to the nearest minute.

See the step-by-step answer guide →