Question

the amount of carbon-14 present in animal bones t years after the animals death is given by $p(t) = p_0 e^{-0.00012097t}$. how old is an ivory tusk that has lost 30% of its carbon-14?

the ivory tusk is \\(\square\\) years old
(round to the nearest integer as needed.)

Explanation:

Step1: Determine remaining carbon-14

Losing 35% means 65% remains: $P(t) = 0.65P_0$

Step2: Substitute into decay formula

$0.65P_0 = P_0e^{-0.00012097t}$ → Cancel $P_0$: $0.65 = e^{-0.00012097t}$

Step3: Take natural log of both sides

$\ln(0.65) = -0.00012097t$

Step4: Solve for t

$t = \frac{\ln(0.65)}{-0.00012097} ≈ \frac{-0.43078}{-0.00012097} ≈ 3561$

Answer:

3561