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.

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 BACC(t) is the BAC after t hoursk 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] [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]