Expert: Paul Klarreich - 2/17/2009

Question
I dont know how to begin. Please guide me. The amount A grams of a given carbon isotope in a dead tree trunk is given by A=A_o (e^-kt) where A_o & k are positive constants, & where the tine is measured in years from the death of the tree. (i)show that A satisfies the equation dA/dt = -kA.(ii)find the value of k if the amount of isotope is halved every 5500 years.(iii)for a particular dead tree trunk the amount of isotope is only 15% of the original amount in the living tree.How long ago did the tree die?Give your answer to the nearest 1000 years. Much appreciated

Answer
Questioner:   Ling
Category:  Advanced Math
Private:  No
 
Subject:  First order differential equations
Question:  I dont know how to begin. Please guide me. The amount A grams of a given carbon isotope in a dead tree trunk is given by A=A_o (e^-kt) where A_o & k are positive constants, & where the tine is measured in years from the death of the tree.

(i)show that A satisfies the equation dA/dt = -kA.

Find  dA/dt, then substitute that and A into the equation.


(ii)find the value of k if the amount of isotope is halved every 5500 years.

A0 is the initial (t = 0) amount.  So halved means A(5500) = A0/2.

Substitute:  

A0/2 = A0 e^-k(5500)

Solve for t.  (you might want a calculator)


(iii) for a particular dead tree trunk the amount of isotope is only 15% of the original amount in the living tree.How long ago did the tree die?Give your answer to the nearest 1000 years. Much appreciated

At this  t,  A = 0.15A0, and you have k from (ii).  Substitute

0.15A0 = A0 e^-kt, where  k = (answer from ii)

Find t.
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Note: You obviously sent me three questions at once. When you do that, you use up my daily quota and block others from asking questions.  That is not considerate of you, and I will reject your other two questions.

It is dishonest and insulting for you to use different names when submitting them.