8 A student is conducting an experiment in a laboratory to investigate how quickly liquids cool to room temperature. A beaker containing a hot liquid at an initial temperature of \(75 ^ { \circ } \mathrm { C }\) cools so that the temperature, \(\theta ^ { \circ } \mathrm { C }\), of the liquid at time \(t\) minutes can be modelled by the equation $$\theta = 5 \left( 4 + \lambda \mathrm { e } ^ { - k t } \right)$$ where \(\lambda\) and \(k\) are constants. After 2 minutes the temperature falls to \(68 ^ { \circ } \mathrm { C }\).
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1. Find the temperature of the liquid after 15 minutes.
   Give your answer to three significant figures.
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2. 1. Find the room temperature of the laboratory, giving a reason for your answer.
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3. (ii) Find the time taken in minutes for the liquid to cool to \(1 ^ { \circ } \mathrm { C }\) above the room temperature of the laboratory.
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4. Explain why the model might need to be changed if the experiment was conducted in a different place.