Scientists can estimate the age of certain ancient items by measuring the proportion of carbon–14, relative to the total carbon content in the item - Leaving Cert Mathematics - Question 3 - 2013

To estimate the age of the item ($t$) given the proportion of carbon–14 as 0.3402, we use the same formula:

$Q = e^{-\frac{0.6931}{5730} t}$

Setting $Q = 0.3402$:

1. Take the natural logarithm: $-\frac{0.6931}{5730} t = \ln(0.3402)$

2. Calculate $\ln(0.3402)$: $\ln(0.3402) \approx -1.078$

3. Rearranging to solve for $t$: $t = -\frac{5730}{0.6931} \cdot \ln(0.3402)$

Substituting values:

$t \approx -\frac{5730}{0.6931} \cdot (-1.078) \approx 8915.68$

Thus, rounding to two significant figures gives:

Estimated age = 8900 years.