Exponential and logarithmic functions

Question

Carbon Dating The age of an ancient artifact can be determined by the amount of radioactive carbon-14 remaining in it. If D0 is the original amount of carbon-14 and D is the amount remaining, then the artifact’s age A (in years) is given by

A=-8267ln (D/D0)

a Find the age of an object if the amount D of carbon-14 that remains in the object is 73% of the original amount D0.

Answer 1

D = D₀e^kt

 

D/D₀ = e^kt

 

kt = ln(D/D₀)

 

k[-8267ln(D/D₀)] = ln(D/D₀)

 

k = -1/8267 ≈ -0.00012

 

D = D₀e^-0.00012t

 

Find t so that D = 0.73D₀

 

0.73D₀ = D₀e^-0.00012t

 

0.73 = e^-0.00012t

 

ln(0.73) = ln(e^-0.00012t)

 

ln(0.73) = -0.00012t

 

t = ln(0.73)/(-0.00012) ≈ 2622.6 years