The half-life of carbon-14 is 5730 years. How long will it take for the number of carbon-14 nuclei in a sample to drop to a quarter of the original number?
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years

11460 years

Explanation:

Let N₀ be the initial amount of carbon present. After one half-life, its value is N₀/2, after two half-lives, it is N₀/4 = N₀/2². After n nalf-lives, it is N = N₀/2ⁿ.

Now, if the value of carbon-14 drops to a quarter of its initial value, then N = N₀/4.

So, N₀/4 = N₀/2ⁿ

1/4 = 1/2ⁿ

1/2² = 1/2ⁿ

Comparing the exponents, n = 2.

So the value of carbon-14 present drops to a quarter of its initial value in 2 half-lives.

Since one-half life equals 5730 years, then two-half-lives is 2 × 5730 years = 11460 years

So, it takes carbon-14 11460 years for the number of nuclei to drop to a quarter of its initial value.

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