Question

find the value of \\( \frac { p ( t ) } { a } \\) for this problem. \\( \frac { p ( t ) } { a } = \\) complete how old is the bone? about 930 years about 1,845 years about 3,720 years about 5,730 years done

Explanation:

Step1: Use the formula for radioactive decay

The formula for radioactive decay is \(P(t)=Ae^{-\lambda t}\), where \(P(t)\) is the amount of the substance at time \(t\), \(A\) is the initial amount, and \(\lambda=\frac{\ln 2}{T}\) (for carbon - 14, \(T = 5730\) years). We know that \(\frac{P(t)}{A}=0.8\). So, \(0.8 = e^{-\frac{\ln 2}{5730}t}\).

Step2: Take the natural logarithm of both sides

Take the natural logarithm of both sides: \(\ln(0.8)=-\frac{\ln 2}{5730}t\).

Step3: Solve for \(t\)

We know that \(\ln(0.8)\approx - 0.2231\) and \(\ln 2\approx0.6931\). Then \(t=\frac{- 5730\times\ln(0.8)}{\ln 2}\). Substitute the values: \(t=\frac{-5730\times(- 0.2231)}{0.6931}\approx1845\).

Answer:

about 1,845 years