Prehistoric cave paintings were discovered in a cave in Fran

Prehistoric cave paintings were discovered in a cave in France. The paint contained 7% of the original carbon-14. Use the exponential decay model for carbon-14, A =A_0 e^-0.00012t, to estimate the age of the paintings. The paintings are approximately years old (Round to the nearest integer.)

Solution

The formula for decay is A = A₀e^-0.000121t where A₀ is the initial quantity of carbon -14 and A is the quantity of carbon -14 after t years. Here, A/A₀ = 7/100 = .07 . Therefore, e^-0.000121t = A/A₀ = 0.07. On taking natural logarithm of both the sides, we hace -0.000121t ln(e) = ln (0.07) or, -0.000121t = -2.659260037 so that t = -2.659260037 / -0.000121 = 21977.35 or 21977 years ( on rounding off to the nearest whole number. [ as ln a^b = b ln a and ln e = 1]. Thus the paintings are approximately 21977 years old.