First lets rearrange the formula to solve for half-life.
The half-life is the time it takes for half of the element to decay.
\(e^{rt} = \frac{1}{2} \\ t = \frac{ln (\frac{1}{2})}{r}\)
So now we need to find "r" which is the rate of decay.
If we rearrange formula to solve for r:
\(P e^{rt} = A \\ \\ r = \frac{ln (\frac{A}{P})}{t}\)
Substituting this into equation for half-life, you get:
\(T = (\frac{ ln (1/2)}{ln (A/P)} ) t = \frac{ ln(1/2)}{ln (200/500)} *(48) = 36.31\)
