2016 RVHS H1 Promo Q11
2016 RVHS H1 Promo Q11
12 marks
The temperature, \(\theta^{\circ}\mathrm{C}\), of the water in a flask, \(t\) minutes after the start of heating is modelled by the equation \(\theta = b + k \ln(2t + 1)\), where \(b\) and \(k\) are constants. Initially the temperature of the water is \(20^{\circ}\mathrm{C}\). After \(2\) minutes of heating, the temperature is \(65^{\circ}\mathrm{C}\).
- Find the values of \(b\) and \(k\).[4]
- Find the time it takes for the water to reach a temperature of \(50^{\circ}\mathrm{C}\).[3]
- Sketch the graph of the temperature of the water against time for \(0 \le t \le 8\), labelling clearly the end points of the graph.[3]
- State, with a reason, whether this model is suitable in the long term.[2]
Solution:
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Answer:(i) \(b=20\), \(k=\dfrac{45}{\ln5}\) (ii) \(0.962\text{ min}\) (3 s.f.) (iii) Increasing, concave-down curve from \((0,20)\) to \((8,99.2)\). (iv) Unsuitable: the model predicts unbounded temperature.
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