Element X decays radioactively with a half life of 13 minutes. If there are 710
grams of Element X, how long, to the nearest tenth of a minute, would it take
the element to decay to 36 grams?

Respuesta :

The half-life of the element is 13 minutes, which means after 13 minutes, any starting amount decays to half. So element X decays with a rate k such that

[tex]\dfrac12 = e^{13k}[/tex]

Solve for k :

[tex]\ln\left(\dfrac12\right) = \ln\left(e^{13k}\right)[/tex]

[tex]-\ln(2) = 13k \ln(e)[/tex]

[tex]-\ln(2) = 13k[/tex]

[tex]\implies k = -\dfrac{\ln(2)}{13}[/tex]

Now, we solve for t such that

[tex]36 = 710e^{kt}[/tex]

[tex]e^{kt} = \dfrac{18}{355}[/tex]

[tex]\ln\left(e^{kt}\right) = \ln\left(\dfrac{18}{355}\right)[/tex]

[tex]kt = \ln\left(\dfrac{18}{355}\right)[/tex]

[tex]-\dfrac{\ln(2)}{13} t = \ln\left(\dfrac{18}{355}\right)[/tex]

[tex]\implies t = -\dfrac{13 \ln\left(\frac{18}{355}\right)}{\ln(2)}[/tex]

[tex]\implies t = -13 \log_2\left(\dfrac{18}{355}\right) \approx \boxed{55.9}[/tex]