Question

The function t relates the age of a plant fossil, in years, to the percentage, c, of carbon-14 remaining in the fossil relative to when it was alive.

t(c)=-3,970(ln c)

Use function t to complete the statements.

A fossil with 47% of its carbon-14 remaining is approximately ___ years old.

A fossil that is 7,000 years old will have approximately ___ of its carbon-14 remaining.

Answer 1

Answer:

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Explanation:

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Answer 2

Applying the logarithmic function, it is found that:

• A fossil with 47% of its carbon-14 remaining is approximately 2997 years old.

• A fossil that is 7,000 years old will have approximately 56.71% of its carbon-14 remaining.

The age of a fossil with a proportion c of carbon 14 remaining is given by:

$t(c) = -3970\ln{c}$

If the fossil has 47% of carbon-14 remaining, we have that $c = 0.47$, and thus:

$t(0.47) = -3970\ln{0.47} = 2997$

For a fossil that is 7000 years old, we have that $t(c) = 7000$, and hence:

$t(c) = -3970\ln{c}$

$\ln{c} = -\frac{3970}{7000}$

$e^{\ln{c}} = e^{-\frac{3970}{7000}}$

$c = 0.5671$

A fossil that is 7,000 years old will have approximately 56.71% of its carbon-14 remaining.

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