Question 473415
The general exponential decay function is: {{{Pt = Po*e^(kt)}}}
Given is that the half-life is 5730. This means that t=5730 when {{{Pt/Po = 1/2}}}
We can solve for k:
{{{e^(5730k) = 1/2}}}
Take natural log of both sides
{{{5730k = ln (1/2)}}}
Divide by 5730 on both sides
{{{k = ln(1/2)/5730}}}
Now substitute this value in for k into general equation, along with fact that {{{Pt/Po = 0.77}}}
Solve for t:
{{{e^(ln(1/2)/5730*t) = 0.77}}}
Take natural log of both sides
{{{ln(1/2)/5730*t = ln 0.77}}}
Multiply by reciprocal -> 5730/ln(1/2)
{{{t = (5730*ln 0.77)/ln(1/2)}}}
Use scientific calculator to estimate answer, round to nearest hundred
t = 2,200 years