# Carbon Dating and a problem that came up in Math class - NOT HOMEWORK HELP

• xenorecor
In summary, my Math teacher and I got into a discussion about carbon dating and exponential decay. The book gave one equation and I (thinking back to chemistry) got another. They both had the same answer (which was great, my teacher was proud) but we could not figure out why. Here it is: 4. If drilling into what was once a lake bottom produces a piece of wood which, according to its mass, would have contained 5 nanograms of carbon-14 when the wood was alive, how much of the carbon would be expected to remain 2 centuries later?

#### xenorecor

So today my Math teacher and I got into a discussion about Carbon Dating and exponential decay. The book gave one equation and I (thinking back to chemistry) got another. They both had the same answer (which was great, my teacher was proud) but we could not figure out why. Here it is:

4. [Suppose that drilling into what was once a lake bottom produces a piece of wood which, according to its mass, would have contained 5 nanograms of carbon-14 when the wood was alive. Use the fact that this radioactive carbon decays continuously at a rate of about 1.2% per century to analyze the sample.]

a. How much of the carbon would be expected to remain 2 centuries later?

Now, I took this and got the following equation: 5 * 10^-9 / 1.012^2

My logic was that 5*10^-9 was the 5 nanograms, and 1.012^2 was the two centuries with the exponential decay. The answer that I got was 4.88 nanograms, which was the same as the book. However, the books equation was:

5(.988)^2 Which is the equation for exponential decay that we learned ( a(b)^2 )

Now for the part we argued a bit on. Why would dividing work in the equation that I got, when you multiply in the book equation? I tried to reason through it and only came up with this:

Because I used 1.012 instead of 0.012, you would have to divide because 1 is the 5 nanograms and 0.012 is the decay. So drawn out I got:

5*10^-9 / (5*10^-9 + 0.012%)^2

Sorry if this is confusing, it is my first shot at trying to work my way through a problem that even my teacher has no answer for. Thanks for any help though!

I'm not sure how much math you have, but here's the explanation.

The general equation for exponential decay is N = N0 * exp(-k*t), where N is the amount remaining, N0 is the original amount, exp(-k) is shorthand for e-k, where e is the Euler constant, k is the decay rate constant (units of 1/time), and t is the elapsed time. If you apply this formula,

N = 5 * exp(-0.012*2) = 4.881429 ng, which is the answer you got.

This formula assumes that the decay occurs continuously during the elapsed time. However, the formula in your book assumes that the decay occurs once per century. The two formulas actually give different answers, if you go to the third decimal place:

N = 5 * (1-0.012)2 = 4.88072 ng

Now, how come both formulas give you similar answers? The reason is that the formula in your book is a simplification. It stems from the fact that exp(-k) can be expressed as a Taylor series as:

exp(-k) = 1 - k/1! + k2/2! - k3/3! + ...

If you drop the higher order terms (exponents larger than 1), exp(-x) approximates 1 - k/1! = 1 - k.

Plugging this back into the original equation, you get

N = N0 * exp(-kt) = N0 * (1 - k) * t

which is the formula in the book.

This problem is similar to the compound interest problems, but in the opposite direction. Instead of decay, in compound interest you see growth. Compound interest can be computed in 2 different ways: if the interest is compounded continuously, the equation is

P = C * exp(rt)

where P is the future value, C is the initial deposit, r is the interest rate, and t is the time.

However, if the interest is compounded once per year, the equation simplifies to

P = C * (1 + r)t

I hope this helps.

I was just going to say that...

Thank you! I didn't understand all of it, but it's starting to make sense. I'll show this to my teacher and see what she can make of it. Thanks again!

## 1. What is carbon dating?

Carbon dating is a method used by scientists to determine the age of organic materials. It works by measuring the amount of carbon-14 present in a sample, which decays at a known rate over time.

## 2. How does carbon dating work?

Carbon dating works by comparing the amount of carbon-14 in a sample with the amount of carbon-14 in the atmosphere. As living organisms absorb carbon-14 from the atmosphere, the ratio of carbon-14 to carbon-12 in their bodies is the same as the ratio in the atmosphere. When an organism dies, it stops absorbing carbon-14 and the amount of carbon-14 begins to decrease due to radioactive decay. By measuring the remaining amount of carbon-14, we can calculate how long ago the organism died.

## 3. What are the limitations of carbon dating?

Carbon dating can only be used to date organic materials, such as wood, bone, and cloth. It is not accurate for dating rocks or other inorganic materials. Additionally, because the amount of carbon-14 in the atmosphere has varied over time, there may be inaccuracies in the dating process. Furthermore, carbon dating cannot accurately date materials older than 50,000 years.

## 4. What problem can arise when using carbon dating in math class?

In math class, students may encounter problems related to calculating the age of a sample using carbon dating. These problems usually involve solving equations or using logarithms to determine the age of a sample based on the amount of carbon-14 remaining.

## 5. How is carbon dating used in other fields of science?

Carbon dating is not only used in archaeology and geology, but also in fields such as biology, forensics, and environmental science. It can be used to determine the age of fossils, track the spread of diseases, and study the effects of human activity on the environment. It is a valuable tool in many areas of scientific research.