Cooling toward ambient temperature

Alice bakes a large pie and removes it from the oven at 1 pm. The temperature reading of the pie using a food thermometer is 200 °C at that instant. She left the pie on the kitchen countertop to cool down. The temperature of the kitchen is kept at a constant 32 °C. The temperature of the pie tt minutes after it is removed from the oven is θ\theta °C. This temperature decreases at a rate proportional to the difference between the temperature of the pie and the temperature of the kitchen. At 1.15 pm, Alice checks the temperature of the pie again and it is 180 °C.

(a) Find an exact expression for θ\theta in terms of tt, simplifying your answer. marks

(b) Sketch a graph of θ\theta against tt. marks

(c) Food safety experts caution that bacteria grow fastest in food within the temperature range of 5 °C to 60 °C, known as the temperature danger zone. Find the latest time to the nearest minute to safely store the pie to avoid this danger zone. marks

[2023/RI/Prelim/I/6]

Total marks: 9

Solution

(a)

Newton's law of cooling

The pie is 200 °C at 1 pm and 180 °C at 1.15 pm; the kitchen is 32 °C. Find an exact expression for θ\theta in terms of tt (minutes).

Explore this idea

Hint

  1. dθdt=−k(θ−32)\frac{d\theta}{dt}=-k(\theta-32), k>0k>0.

  2. Separate and integrate.

  3. Use the two readings.

Answer · (a)

θ=32+168(37/42)t/15\theta=32+168(37/42)^{t/15} °C; tt minutes after 1 pm.

Explore the idea

The cooling pie

Each dash shows the slope the differential equation gives at that point. Change the starting value and the solution follows the dashes.

Starting value of θ at t = 0200
Read the solution at t15
Try an example
θ at t = 15
179.792
Rate dθ/dt there
-1.249
  • Solution that starts at the purple dot
  • Closed-form answer
Key idea

Whatever the start, the solution moves towards the room temperature 32 °C.

Ideas to try
  1. Move the starting temperature below 32: the pie warms up instead.
  2. Read the temperature at t=15t=15 minutes: it should be 180 °C.
(b)

Sketching

Sketch a graph of θ\theta against tt.

Hint

  1. Starts at (0,200)(0,200), decreasing, convex, with horizontal asymptote θ=32\theta=32.

Cooling pie
Cooling pieScroll across to see the full diagram.

Answer · (b)

Decreases from 200, convex, toward θ=32\theta=32.

(c)

Threshold time

The danger zone is 5 °C to 60 °C. Find the latest time (nearest minute) to store the pie safely.

Hint

  1. Solve θ=60\theta=60.

Answer · (c)

Latest rounded time 4:32 pm.

Under the exercise rule, store before the 60°C crossing (t≈212.04 min).