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

The estimated age of the humanoid skull, based on the remaining 2% of original carbon-14 and a half-life of 5730 years, is approximately 31,607 years.

Step-by-step explanation:

Estimating Fossil Age Using Carbon Dating

Carbon dating is a radiometric dating method used to determine the age of carbon-containing materials. When an organism dies, it stops exchanging carbon with the atmosphere, and the carbon-14 (C-14) it contains begins to decay at a known rate, with a half-life of 5730 years. To estimate the age of a fossil or an artifact like the humanoid skull found alongside the remains of a campfire, scientists measure the remaining carbon-14 compared to what would be found in a living organism.

In the given scenario, only 2% of the original C-14 remains in the burnt wood, meaning multiple half-lives have passed. To calculate the age of the skull, which is assumed to be the same as the campfire, the following formula is used:

n = - (log(Total percent of C-14 remaining) / log(2))

where n is the number of half-lives that have elapsed.

Substituting the given values gives:

n = - (log(0.02) / log(2)) ≈ 5.644

Multiplying the number of half-lives by the length of one half-life, we get:

Age = n × half-life = 5.644 × 5600 years ≈ 31607 years

Therefore, the estimated age of the skull is approximately 31,607 years.

Answer 2

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)^{(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.