SOLUTION: The​ half-life of​ carbon-14 is 5600 years. If a piece of charcoal made from the wood of a tree shows only 73​% of the​ carbon-14 expected in living​

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Question 1088726: The​ half-life of​ carbon-14 is 5600 years. If a piece of charcoal made from the wood of a tree shows only 73​% of the​ carbon-14 expected in living​ matter, when did the tree​ die?
The tree died about ______ years ago?
​(Do not round until the final answer. Then round to the nearest whole​ number.)
Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!
formula is:
F+=+P%2Ae%5E%28kt%29 where:
F = future amount
P = present amount
k+= constant of proportion.
t = time
half life of carbon is given by the equation:
.5+=+e%5E%285600k%29 because:
half life of carbon is 5600 years.
solve for k and then you can solve the problem.
take log of both sides of equation of .5+=+e%5E%285600k%29 to get:
log%28.5%29+=+log%28e%5E%285600k%29%29
this becomes:
log%28.5%29+=+5600k+%2A+log%28e%29
divide both sides of this equation by 5600%2Alog%28e%29 to get:
k+=+log%28.5%29%2F%285600%2Alog%28e%29%29
solve for k+to get:
k+=+-.301029996+%2F+%285600%2A.434294482%29 which becomes:
k+=+-.000123776+

if the tree has only 73%=0.73 of the carbon left, then your equation of:
F+=+P+%2A+e%5E%28kt%29 becomes:
.73+=+1%2Ae%5E%28k%2At%29 which becomes:
.73+=+e%5E%28-.000123776%2At%29
take the log of both sides of this equation to get:
log%28.73%29+=+log%28e%5E%28-.000123776%2At%29%29 which becomes:
log%28.73%29+=+-.000123776%2At%2Alog%28e%29
divide both sides of this equation by -.000123776%2Alog%28e%29 to get:
t+=+log%28.73%29+%2F+%28-.000123776%2Alog%28e%29%29
solve for t to get:
t+=+2542.58 years
Then round to the nearest whole number:t+=+2543 years