Differential equation — Coffee cooling with sketch

Question

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 .

This practice question was inspired by Question 12(b) in the 2021 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: . At , , so .
Simplify the previous line carefully, keeping exact values where possible.At , : .
Finish the calculation, then check that the result meets the question’s conditions.Set : minutes.
State the final answer clearly in the form the question requested.(ii) Sketch: , strictly decreasing, concave up (cooling slows as approaches ), horizontal asymptote as . Passes through and about .

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

Verification: (i) About minutes. (ii) Decreasing toward with the listed features.

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