An archeologist in Turkey discovers a spear head that contai

An archeologist in Turkey discovers a spear head that contains 27% of its original amount of C-14 N = N_0e^-kt N_0 = initial amount of C-14 (at time t = 0) N = amount of C-14 at time t k = 0.0001 t = time, in years Find the age of the spear head to the nearest year.

Solution

N = N₀ e ^-kt . Here, N = 27 % of N₀ = 27/100(N₀ ) , k = 0.0001, then 27/100N₀ = N₀ e^-0.001t . Therefore, 0.27 = e^-0.001t .On taking natural logarithms of both the sides, we get - 0.001t ln e = ln 0.27 ( log a^b = b log a), or,

  - 0.001t = -1.30933332 ( as ln e = 1). Therefore, t = 1.30933332/ 0.0001 = 13093.33 years = 13093 years on rounding off to the nearest year.