Question

The radioactive element carbon-14 has a half-life of 5750 years. A scientist determined that the bones from a mastodon

had lost 62.9% of their carbon-14. How old were the bones at the time they were discovered?
The bones were about years old.
(Round to the nearest integer as needed.)

Answer 1

The mastodon bones were approximately 2485 years old when discovered, based on 62.9% remaining carbon-14 and a half-life of 5750 years.

To determine the age of the mastodon bones, we can use the formula for exponential decay:

N(t) = N₀ * (1/2)^(t/T),

where N(t) is the remaining amount of carbon-14 at time t, N₀ is the initial amount of carbon-14, and T is the half-life of carbon-14.

In this case, 62.9% of carbon-14 remains, so N(t) = 0.629, and T = 5750 years. Substituting these values into the formula, we get:

0.629 = (1/2)^(t/5750).

To solve for t, take the natural logarithm (ln) of both sides:

ln(0.629) = t/5750.

Now, solve for t:

t = 5750 * ln(0.629).

Calculating this, we find t ≈ 2485 years. Therefore, the mastodon bones were about 2485 years old when they were discovered. Rounding to the nearest integer, the bones were approximately 2485 years old.