Write an equation modeling this situation

Question

A radioactive isotope has a half life of 1,920 years let f(t) be the amount of isotope, in milligrams, that is present after t years. If the sample originally contained 21 milligrams, write an equation modeling this situation.

F(t)=

Answer 1

We can model this situation with an exponential decay function.

f(t) = Pe^rt

f(t) = amount of isotope

P = starting amount

r = growth rate (our "growth" rate should be negative since it is decreasing over time)

t = time

Let's pretend there is 1 mg of our isotope. That means in 1920 years I will have 1/2 mg. Using these numbers I can plug into my equation and solve for my r value.

(1/2) = (1)e^r(1920)

take the natural log of both sides to bring the exponent down and to cancel out the e

ln(1/2) = ln(e^r(1920))

ln(1/2) = r(1920)

ln(1/2)/1920 = r

r = -.000361

Now our function looks like this...

f(t) = Pe^(-.000361)t

All you have to do now is plug in 21 for P

f(t) = 21e^(-.000361)t

Hope this helps. Please feel free to reach out if you need me to clarify anything :)

Answer 2

1/2 = e^1920r

take natural logs of both sides

1920r = ln.5

r = ln.5/1920 = -3.61 x 10^-4

= - .000361

r = -.0361% mg per year decay rate

F(t) = 21e^.-.000361t

or

F(t) = 21/e^.000361t