Question

(1 point) Carbon dating is often used to determine the age of a fossil. For example, a humanoid skull was found in a cave in South Africa along with the remains of a campfire. Archaeologists believe the age of the skull to be the same age as the campfire. It is determined that only 2% of the original amount of carbon-14 remains in the burnt wood of the campfire. Estimate the age of the skull if the half-life of carbon-14 is about 5600 years.

Answer 1

Answer:

The estimated age of the skull is 39118 years.

Step-by-step explanation:

The amount of the substance after t years is given by:

$A(t) = A(0)(1-r)^t$

In which A(0) is the initial amount, and r is the decay rate.

The half-life of carbon-14 is about 5600 years.

This means that $A(5600) = 0.5A(0)$. We use this to find r, or 1 - r, to replace in the equation. So

$A(t) = A(0)(1-r)^t$

$0.5A(0) = A(0)(1-r)^{5600}$

$(1-r)^{5600} = 0.5$

$\sqrt[5600]{(1-r)^{5600}} = \sqrt[5600]{0.5}$

$1 - r = (0.5)^{\frac{1}{5600}}$

$1 - r = 0.9999$

So

$A(t) = A(0)(0.9999)^t$

Only 2% of the original amount of carbon-14 remains in the burnt wood of the campfire.

This is t for which $A(t) = 0.02A(0)$. So

$A(t) = A(0)(0.9999)^t$

$0.02A(0) = A(0)(0.9999)^t$

$(0.9999)^t = 0.02$

$\log{(0.9999)^t} = \log{0.02}$

$t\log{0.9999} = \log{0.02}$

$t = \frac{\log{0.02}}{\log{0.9999}}$

$t = 39118$

The estimated age of the skull is 39118 years.