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zloy xaker [14]
3 years ago
6

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.
Mathematics
1 answer:
nasty-shy [4]3 years ago
7 0

Answer:

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

Step-by-step 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}\\\frac{A_0}{2} =A_0\,e^{-k\,*22}\\\frac{A_0}{A_0*2} =e^{-k\,*22}\\\frac{1}{2} =e^{-k\,*22}\\ln(\frac{1}{2})=-k\,*22\\ k=-\frac{ln(\frac{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}\\\frac{0.9*A_0}{A_0} =e^{-0.0315\,t}\\0.9 =e^{-0.0315\,t}\\ln(0.9)=-0.0315\,t\\ t=-\frac{ln(0.9)}{0.0315} \\t=3.3447\,hours

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