Find the expression for the mass

Question

The rate of decay in the mass, M, of a radioactive substance is given by the differential equation , where k is a positive constant. If the initial mass was 200g, then find the expression for the mass, M, at any time t.



Question options:

1)	M = 200ln(kt)
2)	M = 2e-kt
3)	M = 200 ekt
4)	M = 200 e-kt

Answer 1

dM/dt = - kM by condition. We separate variables and then integrate: dM∫/M = - kdt; ∫dm/M = ∫- kdt; lnM = - kt + lnC;

From here M = Ce^{- kt}; if t= 0 then M = 200 g, So 200 = Ce⁰ = C, C = 200. And finally we get

M = 200e^-k.

^{Answer: 4)}