Question

The half-life of radium is 1600 years. If the initial amount is
q0 milligrams,
then the quantity
q(t)
remaining after t years is given by
q(t) = q02kt.
Find k.

Answer 1

So the equation is\[
q(t) = q_02^{kt}
\]

Answer 2

Now, the quantity is going to be halved each time so we know \(k\) is going to be negative.
\[
q(t)=q_0\left(\frac{1}{2}\right)^{-kt}
\]

Answer 3

We also know: \[
q(1600) = \frac{q_0}{2} = q\left(\frac{1}{2}\right)^{1} \]Since it is the half-life as well as\[
q(1600)=q_0\left(\frac{1}{2}\right)^{-k(1600)}\]See the pattern?\[
1=-k(1600)
\]

Answer 4

Solving for \(k\) gives us: \[
k = -\frac{1}{1600}
\]

Answer 5

@knbarwrgwddsg get it?