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## Main Question or Discussion Point

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!!

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!!