When was the last eruption of the volcano?

Question

The Lualailua Hills Quadrangle of the East Maui volcano on the island of Maui in Hawaii is no loner active. To find out the date of the last eruption scientists conducted a chemical analysis of samples from the volcano area. The samples contained approx 62.31% of its original carbon-14. Whe,n to the nearest year was the last eruption of the volcano? (use 5715 years for the half life of carbon-14)

Answer 1

This one is tricky, cause you have to use the half-life to find the decay constant as follows.

 

First we know that this equation describes the situation:

 

       R = R₀e^-(t/λ), where R is the decay rate at time t and R₀ is the decay rate at the time of eruption.  λ is the decay constant which we have to figure using the half life.  To do this we make R = R₀/2 and plug it in, getting 1/2 = e^-τ/λ with the greek tau in the numerator equal to the half life.  Here is where it gets tricky:

 

                         1/2 = 1/e^τ/λ and if you take the reciprocal you get

 

                            2 = e^τ/λ

 

Now take ln of both sides and you get ln 2 = τ/λ and using the info from the problem we see that the decay constant is 5715/(ln 2) = 8245.

 

Now we are in a position to solve by graphing y = e^-t/8245.  Using a graphing calculator we find that when y = 0.62,  t = 3900, so the eruption took place that number of years prior to 2015.  As a check when y = 0.5 (the half life amount) t = 5700 as it must