After alcohol is fully absorbed into the body, it is metabolized with a half-life of about 1.5 hours. Suppose you have had three alcoholic drinks and an hour later, at midnight, your blood alcohol concentration (BAC) is 0.5 mg/mL. (a) Find an exponential decay model for your BAC t hours after midnight.

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wifilethbridge wifilethbridge · Answer 1 · 2019-09-12T07:43:41+02:00

Answer:

[tex]C(t)=(0.5)e^{-\frac{ln 2}{1.5}t}[/tex]

Step-by-step explanation:

Given :

After alcohol is fully absorbed into the body, it is metabolized with a half-life of about 1.5 hours.

Your blood alcohol concentration (BAC) is 0.5 mg/mL.

To Find : Find an exponential decay model for your BAC t hours after midnight.

Solution:

General form of decay model : [tex]C(t)=C_oe^{-kt}[/tex]

Where [tex]C_0[/tex] is the initial BAC

C(t) is the BAC after t hours

k is the decay constant

Now we are given that After alcohol is fully absorbed into the body, it is metabolized with a half-life of about 1.5 hours.

Formula : [tex]t_{\frac{1}{2}} =\frac{ln 2}{k}[/tex]

[tex]1.5=\frac{ln 2}{k}[/tex]

[tex]k=\frac{ln 2}{1.5}[/tex]

Your blood alcohol concentration (BAC) is 0.5 mg/mL..

So, [tex]C_0=0.5[/tex]

So, [tex]C(t)=(0.5)e^{-\frac{ln 2}{1.5}t}[/tex]

Hence an exponential decay model for your BAC t hours after midnight is [tex]C(t)=(0.5)e^{-\frac{ln 2}{1.5}t}[/tex]