Carbon dating is used to estimate the age of an ancient human skull. Let \(f(t)\) be the proportion of original \({ }^{14} C\) atoms remaining in the scull after \(t\) years of radioactive decay. Since \({ }^{14} C\) has a half life of 5700 years we have \(f(0)=1\) and \(f(5700)=0.5\).

(a) Sketch the graph of \(f(t)\) versus \(t\) in the domain \(0 \leq t \leq 20000\). Label at least two points of your plot and be sure to label the axes.

(b) Write an expression for \(f(t)\) in terms of \(t\) and other numerical constants such as \(\ln 2, \sin 5, e^{3}\), and \(1 / 5700\). (Note: Not all of these constants need appear in your answer!)

(c) Suppose that only \(15 \%\) of the original \({ }^{14} C\) is found to remain in the skull. Derive from your previous answer, an expression for the estimated age of the skull.