6. A radioactive isotope decays in such a way that the rate of change of the number \(N\) of radioactive atoms present after \(t\) days, is proportional to \(N\).
- Write down a differential equation relating \(N\) and \(t\).
- Show that the general solution may be written as \(N = A \mathrm { e } ^ { - k t }\), where \(A\) and \(k\) are positive constants.
Initially the number of radioactive atoms present is \(7 \times 10 ^ { 18 }\) and 8 days later the number present is \(3 \times 10 ^ { 17 }\).
- Find the value of \(k\).
- Find the number of radioactive atoms present after a further 8 days.