A piece of charcoal is found to contain 30% of the carbon 14 that it originally had. when did the tree die from which the charcoal came? use 5600 years as the half-life of carbon 14.

A (t) = Aoe^kt k<0

Using the equation of half-life, the given is 0.5 = e^ (5600k)

in order to solve for k

we then get the log for both sides

log (0.5) = log (e^ (5600k))

then we simplify

log (0.5) = 5600k * log (e)

k = log (0.5) / (5600*log (e))

k = - 0.301029996 / (5600 * (.434294482)

k = - 0.000123776

Back to the original equation, we have

0.3 = 1*e^ (k*t)

where we substitute for k

0.3 = e^ (-.000123776*t)

and then get the log for each

log (0.3) = log (e^ (-.000123776*t))

and then simplify

log (0.3) = -.000123776*t*log (e)

and then solve for t

t = log (0.3) / (-.000123776*log (e))

t = 9727 years

That is when the tree died.