Anonymous
Anonymous asked in 科學及數學數學 · 1 decade ago

Radiocarbon dating question.

When a living organism dies, its carbon-14 decays. The half-life of carbon dating is 5730 years.

a. Write an equation M(t) expressing the amount of carbon-14 left at time t.

b. The skeleton of mastodon has lost 58% of its original carbon-14. When did the mastodon die?

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  • Anonymous
    1 decade ago
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    Q: When a living organism dies, its carbon-14 decays. The half-life

    of carbon dating is 5730 years.

    a. Write an equation M(t) expressing the amount of carbon-14 left

    at time t.

    b. The skeleton of mastodon has lost 58% of its original carbon-14.

    When did the mastodon die?

    Sol:

    ( a )

    Suppose M0 is the time that living organism dies

    k is constant number and t is the time after organism dies.

    M(t) = M0e-kt

    ( b )

    The half-life of carbon dating is 5730 years

    M(5730) = M0e-5730k = M0 * 50%

    M0e-5730k = M0/2

    e-5730k = 1/2

    ㏑e-5730k = ㏑1/2

    - 5730k = - ㏑2

    k = (㏑2)/5730

    The skeleton of mastodon has lost 58%

    1 - 58% = 42%

    M(t) = M0e-kt = M0 * 42%

    e-kt = 42%

    e-t(㏑2)/5730 = 0.42

    ㏑e-t(㏑2)/5730 = ㏑0.42

    - t(㏑2)/5730 = ㏑0.42

    t㏑2 = - 5730㏑0.42

    t = - 5730㏑0.42/㏑2

    t ≒ 7171.32 ( to 2 d.p. )

    Ans: ( a ) M(t) = M0e-kt ( b ) about 7171.32 years ago

    Source(s): 數學小頭腦
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