Answer:
[tex]t \approx 36.1\,min[/tex]
Step-by-step explanation:
The time constant for the isotope decay is:
[tex]\tau = \frac{8\min}{\ln 2}[/tex]
[tex]\tau \approx 11.542\,min[/tex]
Now, the decay of the isotope is modelled after the following expression:
[tex]m (t) = m_{o}\cdot e^{-\frac{t}{\tau} }[/tex]
The time is now cleared with some algebraic handling:
[tex]\frac{m(t)}{m_{o}} = e^{-\frac{t}{\tau} }[/tex]
[tex]t = -\tau \cdot \ln \frac{m(t)}{m_{o}}[/tex]
Finally, the time need for the element X to decay to 43 grams is:
[tex]t = - (11.542\,min)\cdot \ln\left(\frac{43\,g}{980\,g} \right)[/tex]
[tex]t \approx 36.1\,min[/tex]
