The Rate of Decay

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.

M = 200ln(kt)

M = 2e^-kt

M = 200e^(kt)

M = 200e^(-kt)

Answer 1

dM / dt = -k M Multiply both sides by dt and divide both sides by M, so

dM = -k M dt

dM /M = -k dt Now get integral from both sides:

Ln M = -k t + C1 So:

M = e^(-kt+C1) this is the same as M = e^(-kt) x e ^ C1. Lets assume e^ C1 is C, so

M = C e^ (-kt) Now plug t = 0 and M = 200, so:

200 = C x 1 meaning C = 200 so the equation will b,

M = 200 e ^ (-kt) which is the forth choice.