Question

The​ half-life of a certain tranquilizer in the bloodstream is 22 hours. How long will it take for the drug to decay to 90​% of the original​ dosage? Use the exponential decay​ model, Upper A equals Upper A 0 e Superscript kt​, to solve.

Answer 1

It will take approximately 3.34 hours for the drug to decay to 90% of the original dosage

Explanation:

As suggested, we use the formula for exponential decay:

`A(t)=A_0\,e^(-k\,t)`

From the given information, the half life of the drug in blood id 22 hours, so that means that it takes that number of hours to go from the initial value
`A_0`, to a final value equal to
`A_0/2`. Using this information we can find the decay rate "k" by solving for this parameter in the formula, and using the natural log function to bring the exponent down:

`A(t)=A_0\,e^(-k\,t)\\(A_0)/(2) =A_0\,e^(-k\,*22)\\(A_0)/(A_0*2) =e^(-k\,*22)\\(1)/(2) =e^(-k\,*22)\\ln((1)/(2))=-k\,*22\\ k=-(ln((1)/(2)))/(22) \\k=0.0315`

Now we use this value for the decay rate "k" to calculate how long it would take to decay to 90% of the original dose;

`A(t)=A_0\,e^(-0.0315\,t)\\0.9*A_0} =A_0\,e^(-0.0315\,t)\\(0.9*A_0)/(A_0) =e^(-0.0315\,t)\\0.9 =e^(-0.0315\,t)\\ln(0.9)=-0.0315\,t\\ t=-(ln(0.9))/(0.0315) \\t=3.3447\,hours`