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 (5 billionths of a gram) 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 that carbon-14 would be expected to remain 1 century later?
2 centuries later? x centuries later?
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To analyze the sample and determine how much carbon-14 would be expected to remain after a certain number of centuries, we need to use the fact that carbon-14 decays continuously at a rate of about 1.2% per century. Here's how you can calculate it step by step:
1. Determine the decay factor: The decay factor represents the fraction of carbon-14 that remains after each century. Since carbon-14 decays at a rate of 1.2% per century, the decay factor would be 1 - (1.2/100) = 0.988.
2. Calculate the remaining carbon-14 after 1 century: Multiply the initial amount of carbon-14 by the decay factor. In this case, the initial amount is 5 nanograms. So after 1 century, the remaining carbon-14 would be 5 * 0.988 = 4.94 nanograms.
3. Calculate the remaining carbon-14 after 2 centuries: Multiply the amount of carbon-14 remaining after 1 century by the decay factor. In this case, the remaining carbon-14 after 1 century is 4.94 nanograms. So after 2 centuries, the remaining carbon-14 would be 4.94 * 0.988 = 4.87 nanograms.
4. Calculate the remaining carbon-14 after x centuries: To find the remaining carbon-14 after x centuries, you can use the formula:
Remaining carbon-14 = Initial amount * (decay factor) ^ x
Replace the initial amount with 5 nanograms and the decay factor with 0.988. For example, if x = 10 centuries, the calculation would be: Remaining carbon-14 = 5 * (0.988) ^ 10 = 4.36 nanograms.
So, to summarize:
- After 1 century: 4.94 nanograms
- After 2 centuries: 4.87 nanograms
- After x centuries: 5 * (0.988) ^ x nanograms