Question

The half-life of carbon-14 is 5600 years. If a piece of charcoal made from the wood of a tree shows only 78% of the carbon-14 expected in living matter, when did the tree die?

Answer 1

2,700 years

Step-by-step explanation:

We were given the following details:

Half-life of carbon-14, T = 5600 years

78% of carbon-14 is left. This indicates that the amount of carbon-14 left is 78% of the initial amount:

`0.78* A_0`

We will calculate the time since when the tree died as shown below:

`\begin{gathered} 0.78* A_0=A_0*(0.5)^{(t)/(T)} \\ T=5600years \\ t=? \\ \text{Substitute the variables into the formula, we have:} \\ 0.78=(0.5)^{(t)/(5600)} \\ \text{Take the natural logarithm of both sides, we have:} \\ ln(0.78)=(t)/(5600)* ln(0.5) \\ \text{Make ''t'' the subject of the formula, we have:} \\ t=5600*(ln(0.78))/(ln(0.5)) \\ t=5600*0.35845397091 \\ t=2007.34223711\approx2007 \\ t=2,007years \end{gathered}`

Therefore, the tree died 2,700 years ago