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).

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)

a) The model for the decay of a particular radioactive isotope can be represented using the following equation:

r(t) = r0 * e^(-λt)

In this equation, r(t) represents the amount of the radioactive isotope present at time t, r0 represents the initial amount of the isotope (at t = 0), e represents the mathematical constant (approximately equal to 2.71828), ^ represents exponentiation, -λ represents the decay rate (with a negative sign), and t represents the time.

b) The corresponding initial-value problem for the model in part (a) would be:

r(0) = r0

This initial condition states that at time t = 0, the amount of the isotope present (r(0)) is equal to the initial amount of the isotope (r0).

a) The model for the decay of a particular radioactive isotope can be written as:

r(t) = r0 * e^(-λt)

Here, r(t) represents the amount of the radioactive isotope present at time t, r0 represents the initial amount of the isotope at t = 0, λ represents the decay rate, and e is the base of the natural logarithm.

b) The corresponding initial-value problem for the model in part (a), considering an initial amount of the isotope r0 at t = 0, is:

r(0) = r0

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