Question 1

Paper 2, calculatorMedium12 marks

A liquid is cooling in a room where the temperature remains at 20 ∘C20\,^\circ\mathrm C The measured temperatures are shown in Figure 1. Temperatures were recorded to the nearest 1 ∘C1\,^\circ\mathrm C

For 0≤t≤150\leq t\leq 15 where tt is measured in minutes and TT is measured in degrees Celsius, the cooling is modelled by

dTdt=−k(T−20),k>0,\frac{dT}{dt}=-k(T-20), \qquad k>0,

where kk is constant.

05101520304050607080t / minT / °C804334
(a)

Write down the measured temperatures at t=0t=0 and t=10t=10 minutes, as shown in Figure 1.

[2]
(b)

Solve the differential equation and use the measurements in Figure 1 to determine kk giving your answer to four significant figures with units.

[4]
(c)

Determine estimates for TT at t=5t=5 and t=10t=10 minutes using Euler’s method with step length h=5h=5 minutes. Give both estimates to three significant figures.

[3]
(d)

Hence, use the solution from part (b) to determine the model’s predicted temperature at t=15t=15 minutes. Give your answer to the nearest 0.1 ∘C0.1\,^\circ\mathrm C and compare it with the measurement in Figure 1.

[3]

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