C3 January 2008 Q5
5. The radioactive decay of a substance is given by\[R = 1000\mathrm{e}^{-ct}, \qquad t \geqslant 0.\]where \(R\) is the number of atoms at time \(t\) years and \(c\) is a positive constant.
(a) Find the number of atoms when the substance started to decay. (1)
It takes 5730 years for half of the substance to decay.
(b) Find the value of \(c\) to 3 significant figures. (4)
(c) Calculate the number of atoms that will be left when \(t = 22\,920\). (2)
(d) In the space provided, sketch the graph of \(R\) against \(t\). (2)
| Scheme | Marks |
|---|---|
| 1000 | B1 |
| (1) |
| Scheme | Marks |
|---|---|
| \(1000\mathrm{e}^{-5730c} = 500\) | M1 |
| \(\mathrm{e}^{-5730c} = \dfrac{1}{2}\) | A1 |
| \(-5730c = \ln\dfrac{1}{2}\) | M1 |
| \(c = 0.000121\) cao | A1 |
| (4) |
| Scheme | Marks |
|---|---|
| \(R = 1000\mathrm{e}^{-22920c} = 62.5\) Accept 62-63 | M1 A1 |
| (2) |
| Scheme | Marks |
|---|---|
| Shape | B1 |
| 1000 | B1 |
| (2) | |
| (9 marks) |