Posted by **Quinn** on Thursday, January 17, 2013 at 10:08pm.

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).

- Differential Equations -
**bobpursley**, Thursday, January 17, 2013 at 10:27pm
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

- Differential Equations -
**Ryan Ward**, Thursday, February 7, 2013 at 12:04am
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)

## Answer This Question

## Related Questions

- Math - D.E.Q. - The half-life of a radioactive isotope is the amount of time it ...
- calculus - Carbon-14 is a radioactive substance produced in the Earth's ...
- Calculus - My teacher wants me to use this formula for half life question C(t...
- chem - a) Potassium-40 undergoes three different modes of radioactive decay (...
- physics - a radioactive sample has a decay rate R of 518 decay/min at time t=7 ...
- chem - Potassium-40 undergoes three different modes of radioactive decay (Beta ...
- ap calculus - suppose that the amount in grams of a radioactive substance ...
- Math - A radioactive substance decays according to the formula Q(t) = Q0e−...
- Math - A radioactive substance decays according to the formula Q(t) = Q0e−...
- Chemistry - How many half lives will it take for a sample of a radioactive ...

More Related Questions