C3 June 2006 Q4
4. A heated metal ball is dropped into a liquid. As the ball cools, its temperature, \(T\,{}^\circ\mathrm{C}\), \(t\) minutes after it enters the liquid, is given by\[T = 400\mathrm{e}^{-0.05t} + 25, \quad t \geqslant 0.\]
(a) Find the temperature of the ball as it enters the liquid. (1)
(b) Find the value of \(t\) for which \(T = 300\), giving your answer to 3 significant figures. (4)
(c) Find the rate at which the temperature of the ball is decreasing at the instant when \(t = 50\). Give your answer in \({}^\circ\mathrm{C}\) per minute to 3 significant figures. (3)
(d) From the equation for temperature \(T\) in terms of \(t\), given above, explain why the temperature of the ball can never fall to \(20\,{}^\circ\mathrm{C}\). (1)
| Scheme | Marks |
|---|---|
| \(425\,{}^\circ\mathrm{C}\) | B1 |
| (1) |
| Scheme | Marks |
|---|---|
| \(300 = 400\mathrm{e}^{-0.05t} + 25 \Rightarrow 400\mathrm{e}^{-0.05t} = 275\) sub. \(T = 300\) and attempt to rearrange to \(\mathrm{e}^{-0.05t} = a\), where \(a \in \mathbb{Q}\) | M1 |
| \(\mathrm{e}^{-0.05t} = \dfrac{275}{400}\) | A1 |
| M1 correct application of logs | M1 |
| \(t = 7.49\) | A1 |
| (4) |
| Scheme | Marks |
|---|---|
| \(\dfrac{\mathrm{d}T}{\mathrm{d}t} = -20\mathrm{e}^{-0.05t}\) (M1 for \(k\mathrm{e}^{-0.05t}\)) | M1 A1 |
| At \(t = 50\), rate of decrease \(= (\pm)\,1.64\,{}^\circ\mathrm{C/min}\) | A1 |
| (3) |
| Scheme | Marks |
|---|---|
| \(T \gt 25\), (since \(\mathrm{e}^{-0.05t} \to 0\) as \(t \to \infty\)) | B1 |
| (1) | |
| (9 marks) |