Question 892641
{{{p=Ie^(-kt)}}}, generalized exponential decay equation
p, how much present at time t starting from initial amount I
I, initial amount when t=0
t, time
k, a constant
e, the Euler number, base of Natural logarithm


First, find k using I=80, p=9, t=3.
{{{ln(p)=ln(Ie^(-kt))}}}
{{{ln(I)+ln(e^(-kt))=ln(p)}}}
{{{ln(I)+(-k)t*1=ln(p)}}}
{{{-kt=ln(p)-ln(I)}}}
{{{k=-(1/t)(ln(p)-ln(I))}}}
{{{highlight(k=-(1/t)(ln(p/I)))}}}
For the value of k:
{{{k=-(1/3)(ln(9/80))}}}
{{{highlight(k=0.728)}}}, using k as a positive value.


Half-Life
Here, {{{p=I/2}}}.
{{{p=I/2=Ie^(-0.728*t)}}}
{{{1/2=e^(-0.728t)}}}
{{{ln(1/2)=-0.728t*1}}}
{{{t=-1/(0.728)ln(1/2)}}}
{{{highlight(t=0.952)}}} in hours.