Question

The half-life of uranium-232 is 72 years. Find the approximate value of k in the formula y = ne^kt for uranium-232. Assume t is in years.

Answer 1

Given formula that represents the quantity of uranium-232 in t years,

$y=ne^{kt}$               ...... (1)

Initially, t=0,

So, the quantity of uranium-232 is,

$y=ne^{0}=n$

Thus, half quantity of initial uranium-232 is $\frac{n}{2}$,

According to the question, half life is 72 years.

Substitute $y=\frac{n}{2}$ and t=72 in equation (1),

  $\frac{n}{2}=ne^{72k}$

  $\frac{1}{2}=e^{72k}$

$0.5=e^{72k}$

Take ln both sides,

$\ln (0.5)=72k \ln e$                     $(\because \ln a^b=b \ln a)$

$\ln(0.5)=72k$                           $(\because \ln e=1)$

        $k=\frac{\ln(0.5)}{72}\approx -0.009627$  (by calculator)

Hence, the approximate value of k would be −0.009627.