Respuesta :
Answer:
It will take 4.84 hours for the drug to decay to 93% of the original dosage.
Step-by-step explanation:
We are given that the half-life of a certain tranquilizer in the bloodstream is 47 hours.
The given exponential model is: [tex]A = A_0 e^{kt}[/tex]
Now, we know that A becomes half after 47 hours which means that;
[tex]A = 0.5 A_0[/tex]
Using this in the above equation we get;
[tex]A = A_0 e^{kt}[/tex]
[tex]0.5 A_0 = A_0 e^{(k\times 47)}[/tex] where t = 47 hours
[tex]\frac{0.5 A_0}{A_0} = e^{(47k)}[/tex]
[tex]0.5 = e^{47k}[/tex]
Taking log on both sides we get;
[tex]ln(0.5) = ln(e^{47k})[/tex]
[tex]ln(0.5) =47k[/tex]
[tex]k = \frac{ln(0.5)}{47}[/tex]
k = -0.015
Now, the time it will take for the drug to decay to 93% of the original dosage is given by;
[tex]0.93 = e^{kt}[/tex] where t is the required time
[tex]0.93 = e^{(-0.015 \times t)}[/tex]
Taking log on both sides we get;
[tex]ln(0.93) = ln(e^{-0.015t})[/tex]
[tex]ln(0.93) =-0.015t[/tex]
[tex]t = \frac{ln(0.93)}{-0.015}[/tex]
t = 4.84 hours
Hence, it will take 4.84 hours for the drug to decay to 93% of the original dosage.
