Answer to Question #280883 in Differential Equations for Ward

Question

Many scientists believe that large asteroid struck the earth that kill off dinosaurs. Fragment of asteroid had found out and contain 86% of its original uranium-238. How old is the fragment of the asteroid? The half-life of uranium-238 is 4.5 billion years

Answer 1

Decay equation

$$N(t)=N_0e^{-\lambda t}$$

The half-life of uranium-238 is 4.5 billion years

$$\dfrac{N_0}{2}=N_0e^{-\lambda (4.5\times10^9)}$$

$$e^{\lambda (4.5\times10^9)}=2$$

$$\lambda=\dfrac{\ln 2}{4.5\times10^9}$$

Then

$$N(t)=N_0e^{-(\ln2/(4.5\times10^9))t}$$

$$N(t)=N_0(2)^{-t/(4.5\times 10^9)}$$

Given $N(t_1)=0.86N_0$

$$0.86N_0=N_0(2)^{-t/(4.5\times 10^9)}$$

$$t_1=-4.5\times10^9\cdot\big(\dfrac{\ln 0.86}{\ln2}\big)$$

$$t_1=979\times10^6 \ years$$

The fragment of the asteroid is 979 million years old.