Question

Enter the exponential decay function described in the situation and answer the question asked. You may find using a graphing calculator helpful in solving this problem.

Carbon-14 is a radioactive isotope of carbon that is used to date fossils. There are about 1.5 atoms of carbon-14 for every trillion atoms of carbon in the atmosphere, which known as 1.5 ppt (parts per trillion). Carbon in a living organism has the same concentration as carbon-14. When an organism dies, the carbon-14 content decays at a rate of 11.4% per millennium (1,000 years). Write the equation for carbon-14 concentration (in ppt) as a function of time (in millennia) and determine how old a fossil must be that has a measured concentration of 0.3 ppt. Round your answer to two decimal places.

The exponential decay function is c (t) = _________ .

The fossil is about ________ millennia old.

Answer 1

Intial concentration of carbon-14 = 1.5 ppt

C-14 decay rate = 11.4% per millennium

this means 11.4 % of intial concentration of C-14 decays per millennium

so after 1 millennium, amount of C-14 left = 11.4% of Intial concentration of carbon-14

= (11.4 X 1.5) /100

after 1 millennium, amount of C-14 left = 0.171ppt

To find amount of C-14 left after 2 millennium, we will use intial conc of C-14 = 0.171 ppt and we will find 11.4% of this intial conc of C-14.

so after 2 millennium, amount of C-14 left = 11.4% of Intial concentration of carbon-14

= (11.4 X 0.171) /100

after 2 millennium, amount of C-14 left = 0.019494ppt

To find amount of C-14 left after 3 millennium, we will use intial conc of C-14 = 0.019494 ppt and we will find 11.4% of this intial conc of C-14.

so after 3 millennium, amount of C-14 left = 11.4% of Intial concentration of carbon-14

= (11.4 X 0.019494) /100

after 3 millennia, amount of C-14 left = 0.02222ppt

similarly , we will find amount of C-14 left after 4,5,6,7,8,9 and 10 millennia.

then , we will plot a graph between  carbon-14 concentration (in ppt) and time (in millennia) using excel and equation of the graph will give the exponential decay function.

time	C-14 conc
(in millennia)	(in ppt)
1	0.171
2	0.019494
3	0.002222
4	0.000253
5	2.89E-05
6	3.29E-06
7	3.75E-07
8	4.28E-08
9	4.88E-09
10	5.56E-10

y = 1.5e^-2.172x is the equation of the graph.

where y-axis is conc of C-14(in ppt) at time 't' and x-axis is time(t) in millennium

so, The exponential decay function is c (t) =1.5e^-2.172t

here 1.5 is the intial conc of C-14 = c(0)

and 2.172 is the rate constant for the decay of C-14 content= k

using the equation of graph we can find the age of fossil whose conc is 0.3 ppt by putting c(t) = 0.3ppt

so, c (t) =1.5e^-2.172t

0.3 ppt = 1.5 e^-2.172t

     e^-2.172t = 0.3/1.5 = 0.2

taking ln of this

-2.172t = ln (0.2) = -1.609

t = -1.609 / (-2.172) = 0.74

The fossil is about 0.74 millennia old.