Question

Model radioactive decay using the notation
t = time (independent variable)
r(t) = amount of particular radioactive isotope present at time t (dependent variable)
-λ = decay rate (parameter)
Note that the minus sign is used so that λ > 0
a) Using this notation, write a model for the decay of a particular radioactive isotope.
b) If the amount of the isotope present at t = 0 is r0, state the corresponding initial-value problem for the model in part (a).

Answer 1

the rate is proportional to r(t) times a constant

dr/dt=r(t)* constant where the constant is negative

and the solution to this first order diff equation is of the form

r(t)=K*e^(- λ)t + C

b. ro=K+C
and at t=inf, r(inf)=0
which implies C is zero, so k=ro.
r(t)=ro*e^- λ t

Answer 2

a) dr(t)/dt = -λr(t)

=>∫[ dr(t)/r(t) ] = ∫[ -λt ]
=> ln |r| = -λt + C
=> r(t) = e^(-λt + C) = e ^ (C - λt)
=> r(t) = e^C * e ^ (-λt)

let C = e^C
=> r(t) = Ce^(-λt)

b) r(0) = r_0

=>r(0) = r_0 = Ce^(-λ(0))
=>r_0 = C

=>r(t) = r_0 * e^(-λt)