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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.
So the equation is\[ q(t) = q_02^{kt} \]
Now, the quantity is going to be halved each time so we know \(k\) is going to be negative. \[ q(t)=q_0\left(\frac{1}{2}\right)^{-kt} \]
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?\[ 1=-k(1600) \]
Solving for \(k\) gives us: \[ k = -\frac{1}{1600} \]