Respuesta :
We have to model the exponential decay of the drug quantity Q.
The time t = 0 will correspond to the moment where the dose has been injected totally.
The value of Q at time t = 0, that is Q(0), then has a value:
[tex]Q(0)=0.2[/tex]Then, the value of Q decays exponentially. The rate is 0.4% per second.
We can start expressing the general model for Q:
[tex]Q(t)=A\cdot b^t[/tex]where A and b are parameters of the model.
We can find A as:
[tex]\begin{gathered} Q(0)=Ab^0 \\ Q(0)=A\cdot1 \\ Q(0)=A \\ A=0.2 \end{gathered}[/tex]Parameter A correspond to the initial value of Q(0), which is 0.2 ml.
Now, we will use the rat of change of Q to find the parameter k.
As Q decays by 0.4% per second, we can write:
[tex]\begin{gathered} Q(t+1)=(1-\frac{0.4}{100})\cdot Q(t) \\ Q(t+1)=(1-0.004)\cdot Q(t) \\ Q(t+1)=0.996\cdot Q(t) \end{gathered}[/tex]Then, we can rearrange it as:
[tex]\begin{gathered} \frac{Q(t+1)}{Q(t)}=0.996 \\ \frac{0.2\cdot b^{t+1}}{0.2\cdot b^t^{}}=0.996 \\ b^{t+1-t}=0.996 \\ b^1=0.996 \\ b=0.996 \end{gathered}[/tex]We then can express the model for Q as:
[tex]Q(t)=0.2\cdot0.996^t[/tex]Answer: Q(t) = 0.2*(0.996)^t
