June 2022 Paper 1 Q14

OCR MEICurrent spec13 marksLogs & ExponentialsModelling

14 Alex places a hot object into iced water and records the temperature \(\theta\,{}^\circ\mathrm{C}\) of the object every minute. The temperature of an object \(t\) minutes after being placed in iced water is modelled by \(\theta = \theta_0\mathrm{e}^{-kt}\) where \(\theta_0\) and \(k\) are constants whose values depend on the characteristics of the object.

The temperature of Alex’s object is \(82\,{}^\circ\mathrm{C}\) when it is placed into the water. After 5 minutes the temperature is \(27\,{}^\circ\mathrm{C}\).

(a) Find the values of \(\theta_0\) and \(k\) that best model the data. [3]
(b) Explain why the model may not be suitable in the long term if Alex does not top up the ice in the water. [1]
(c) Show that the model with the values found in part (a) can be written as \(\ln\theta = a - bt\) where \(a\) and \(b\) are constants to be determined. [2]

Ben places a different object into iced water at the same time as Alex. The model for Ben’s object is \(\ln\theta = 3.4 - 0.08t\).

(d) Determine each of the following:
  • the initial temperature of Ben’s object
  • the rate at which Ben’s object is cooling initially.
[4]
(e) According to the models, there is a time at which both objects have the same temperature.
Find this time and the corresponding temperature. [3]
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