9. A scientist is using carbon-14 dating to determine the age of some wooden items.
The equation for carbon-14 dating an item is given by
$$N = k \lambda ^ { t }$$
where
- \(N\) grams is the amount of carbon-14 currently present in the item
- \(k\) grams was the initial amount of carbon-14 present in the item
- \(t\) is the number of years since the item was made
- \(\lambda\) is a constant, with \(0 < \lambda < 1\)
- Sketch the graph of \(N\) against \(t\) for \(k = 1\)
Given that it takes 5700 years for the amount of carbon-14 to reduce to half its initial value,
show that the value of the constant \(\lambda\) is 0.999878 to 6 decimal places.
Given that Item \(A\)
- is known to have had 15 grams of carbon-14 present initially
- is thought to be 3250 years old
- calculate, to 3 significant figures, how much carbon-14 the equation predicts is currently in Item \(A\).
Item \(B\) is known to have initially had 25 grams of carbon-14 present, but only 18 grams now remain.Use algebra to calculate the age of Item \(B\) to the nearest 100 years.