Answer:
There are [tex]9.537\times 10^{-23}[/tex] kilograms of radioactive material after 300 seconds.
Explanation:
From Physics we know that radioactive materials decay at exponential rate, whose differential equation is:
[tex]\frac{dm}{dt} = -\lambda\cdot m[/tex] (1)
Where:
[tex]\frac{dm}{dt}[/tex] - Rate of change of the mass of the radioactive material, measured in kilograms per second.
[tex]m[/tex] - Current mass of the radioactive material, measured in kilograms.
[tex]\lambda[/tex] - Decay constant, measured in [tex]\frac{1}{s}[/tex].
The solution of the differential equation is:
[tex]m(t) = m_{o}\cdot e^{-\lambda\cdot t}[/tex] (2)
Where:
[tex]m_{o}[/tex] - Initial mass of the radioactive material, measured in kilograms.
[tex]t[/tex] - Time, measured in seconds.
If we know that [tex]m_{o} = 20\,kg[/tex], [tex]\lambda = 0.179\,\frac{1}{s}[/tex] and [tex]t = 300\,s[/tex], then the initial mass of the radioactive material is:
[tex]m(t) = (20\,kg)\cdot e^{-\left(0.179\,\frac{1}{s} \right)\cdot (300\,s)}[/tex]
[tex]m(t) \approx 9.537\times 10^{-23}\,kg[/tex]
There are [tex]9.537\times 10^{-23}[/tex] kilograms of radioactive material after 300 seconds.
