Differential equation — New initial drop and target

Question

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.

This practice question was inspired by Question 12(d) in the 2022 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: separate or integrate the differential equation, include the constant, then use the initial condition.(i) Separate and integrate: . At , , so .
Simplify the previous line carefully, keeping exact values where possible., .
Simplify the previous line carefully, keeping exact values where possible.Hence .
Simplify the previous line carefully, keeping exact values where possible.(ii) Set : .
Finish the calculation, then check that the result meets the question’s conditions. minutes.
State the final answer clearly in the form the question requested.Answer: (i) as shown; (ii) minutes.

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

Verification: Dependent parts: calibrate the model, then use it to solve for time.

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