Anonymous

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

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