Half life of carbon-14: age if 1/10 original amt remains

[Emma1703]

Radiocarbon dating can be used to estimate the age of material that comes from a formerly living organism, such as an animal or a plant. When the organism dies, the amount of carbon-14 in its remains decays exponentially. Suppose that C(t) is the amount of carbon-14 in an organism at time t (in years), where t = 0 corresponds to the time of the death of the organism. Assume that the amount of carbon-14 is modelledby the exponential decay function C(t) = C₀e^kt (t ≥ 0), where C₀ is the initial amount of carbon-14 and k is a constant.

The half-life (or halving period) of a radioactive substance like carbon-14 is the time that it takes for the amount of the substance to decrease to half of its original level.

(a) Given that the half-life of carbon-14 is 5730 years, find the value of the constant k, correct to three significant figures. [3]

(b) Suppose that some organic remains have been found where the amount of carbon-14 is known to have decreased to 1/10 of the level that was present immediately after the death of the organism. How much time has elapsed since the death of this organism? Give your answer to the nearest whole number of years.

 

[Deleted member 4993]

Radiocarbon dating can be used to estimate the age of material thatcomes from a formerly living organism, such as an animal or a plant.When the organism dies, the amount of carbon-14 in its remains decaysexponentially. Suppose that C(t) is the amount of carbon-14 in anorganism at time t (in years), where t = 0 corresponds to the time of thedeath of the organism. Assume that the amount of carbon-14 is modelledby the exponential decay function
C(t) = C0ekt (t ≥ 0),
where C0 is the initial amount of carbon-14 and k is a constant.
The half-life (or halving period) of a radioactive substance like carbon-14is the time that it takes for the amount of the substance to decrease tohalf of its original level.

2. (a) Given that the half-life of carbon-14 is 5730 years, find the value of
   the constant k, correct to three significant figures. [3]
3. (b) Suppose that some organic remains have been found where the
   amount of carbon-14 is known to have decreased to 1/10 of the level
   that was present immediately after the death of the organism. How
   much time has elapsed since the death of this organism? Give your
   answer to the nearest whole number of years.

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[stapel]

Radiocarbon dating can be used to estimate the age of material that comes from a formerly living organism, such as an animal or a plant. When the organism dies, the amount of carbon-14 in its remains decays exponentially. Suppose that C(t) is the amount of carbon-14 in an organism at time t (in years), where t = 0 corresponds to the time of the death of the organism. Assume that the amount of carbon-14 is modelledby the exponential decay function C(t) = C₀e^kt (t ≥ 0), where C₀ is the initial amount of carbon-14 and k is a constant.

The half-life (or halving period) of a radioactive substance like carbon-14 is the time that it takes for the amount of the substance to decrease to half of its original level.

(a) Given that the half-life of carbon-14 is 5730 years, find the value of the constant k, correct to three significant figures.

They've given you an equation, a time t when the original (100%) value is halved (50%), and asked you to find the decay constant k. So plug the given information into the given equation, and solve for the remaining variable.

Where are you stuck? Please show all of your steps so far. Thank you!

 

[Emma1703]

I think I have worked out the first part and have the answer as K=-0.000121 to 3 significant figures. Not really sure where to start for part b :sad:

 

[stapel]

I think I have worked out the first part and have the answer as K=-0.000121 to 3 significant figures. Not really sure where to start for part b :sad:

You were given an equation with k and t. You've solved for k. You're now given that C = (1/10)C₀. You've been asked for the value of t. So plug all the knowns and givens into the given equation, and solve for the remaining variable.

 

[SabreFrenzy]

Heres my working,

Here's my workings. I thought I had it but by check shows me I have gone adrift. could anyone give me a pointer?

 

[SabreFrenzy]

Heres my working,

Hopefully I have attached my working. I thought I had it but then my check showed me I was wrong... any ideas?

 

[stapel]

\(\displaystyle 5730\, \times\, \dfrac{1}{10}\, =\, 573\, \mbox{ years}\)

You were given that 5,730 years was the half-life of the element; that is, that it takes 5,730 years for the original amount to decay to only half of that original amount. What are you doing here? You appear to be multiplying the half-life time-period by the amount left over (one-tenth of the original amount), and declaring (at the beginning) that the number of years for one-tenth of the amount to be left is one-tenth of the half-life.

If it takes 5,730 years for 100% to go down to 50%, how are you getting the it takes only 573 years from 100% to go all the way down to 10%? :shock:

 

[Ishuda]

I think I have worked out the first part and have the answer as K=-0.000121 to 3 significant figures. Not really sure where to start for part b :sad:

Personally, having obtained
k = -\(\displaystyle \frac{ln(2)}{5730}\),
I would have rewritten the equation as
C(t) = C₀ \(\displaystyle 2^{-\frac{t}{5730}}\)
and then, for the second part, one would have
\(\displaystyle \frac{1}{10}\, C_0\, =\, C_0\, 2^{-\frac{t}{5730}}\)
and could then solve for t.

 

[Zohaibali]

What is the value of C(t) and Co??

They've given you an equation, a time t when the original (100%) value is halved (50%), and asked you to find the decay constant k. So plug the given information into the given equation, and solve for the remaining variable.

Where are you stuck? Please show all of your steps so far. Thank you!

What is the value of C(t) and Co??

 

[stapel]

What is the value of C(t) and Co??

Working from what was provided earlier in this thread, what values did you get? How?

Please show all of your work and reasoning so far. Thank you!

 

[Zohaibali]

I don't what is the value of Co and C(t)? How to fetch these two values?

Working from what was provided earlier in this thread, what values did you get? How?

Please show all of your work and reasoning so far. Thank you!

I don't what is the value of Co and C(t)? How to fetch these two values?

 

[stapel]

I don't what is the value of Co and C(t)? How to fetch these two values?

You could start by using what you've learned in class, and then using what was provided earlier in this thread.

Before re-re-posting your question to the other student's thread, please do some work of your own. If you are unable to find the solution, then reply showing all of your work. (If you are unable even to get started, then you need more help than we can here provide.)

Thank you!