I have this system in maple and I can't get it work write:
sist:=diff(x(t), t) =-k*x(t), x(0)=x0;
Can someone help me get the desintegration constant k if the half-time is t1/2=700*10^6
Thanks
I have this system in maple and I can't get it work write:
sist:=diff(x(t), t) =-k*x(t), x(0)=x0;
Can someone help me get the desintegration constant k if the half-time is t1/2=700*10^6
Thanks
The solution of the differential equation is $x(t)=x_0e^{-kt}.$
To say that the half-life is $h$ is to say that $x(h)=\frac{x_0}{2}$. Substituting in the formula for $x(t)$, we get $x_0e^{-kh}=\frac{x_0}{2}.$
Cancel the $x_0$, and take reciprocals (this part is not strictly necessary). We get $e^{kh}=2.$ Take the natural logarithm, which we will call $\ln$. We get $kh=\ln 2\quad\text{and therefore} \quad k=\frac{\ln 2}{h}.$ Now use the given value $700\times 10^6$ of the half-life. I get that $k$ is approximately $9.902\times 10^{-10}$.