The average blood alcohol concentration (BAC) of eight male subjects was measured after consumption of 15 mL of ethanol (corresponding to one alcoholic drink). The resulting data were modeled by the concentration function C(t) = 0.0225te−0.0467t where t is measured in minutes after consumption and C is measured in mg/mL. (Round your answers to five decimal places.) (a) How rapidly was the BAC increasing after 9 minutes? (mg/mL)/min (b) How rapidly was it decreasing half an hour later (t = 30)? (mg/mL)/min

Since you need to determine how rapidly the concentrations will vary (increase or decrease) at different times, this is the instantaneous speed or rate of change, you need to find the first derivative of C(t), C'(t)

The next steps show the use of the rule of chain to find C'(t):

[tex]C(t) = 0.0225te^{-0.0467t}\\ \\ C'(t)=0.0225[e^{-0.0467t}-0.0467te^{-0.0467t}]\\ \\ C'(t)=0.0225e^{-0.0467t}-0.00105075e^{-0.00467t}t[/tex]

Now that you have the expression for the speed at which the concentration changes, you just need to substitute the values.

(a) How rapidly was the BAC increasing after 9 minutes? (mg/mL)/min

[tex]C'(9)=0.0225e^{-0.0467(9)}-0.00105075(9)(e^{-0.0467(9)})}\\ \\ C'(t)=0.00857mg/min[/tex]

(b) How rapidly was it decreasing half an hour later (t = 30)? (mg/mL)/min

[tex]C'(30)=0.0225e^{-0.0467(30)}-0.00105075(30)(e^{-0.0467(30)})}\\ \\ C'(t)=-0.00222mg/min[/tex]