The attending physician in an emergency room treats an unconscious patient suspected of a drug overdose. The physician does not know the initial concentration $A_{0}$ of the drug in the bloodstream at the time of injection. However, the physician knows that after $3 \mathrm{hr}$, the drug concentration in the blood is $0.69 \mu \mathrm{g} / \mathrm{dL}$ and after $4 \mathrm{hr}$, the concentration is $0.655 \mu \mathrm{g} / \mathrm{dL}$. The model $A(t)=A_{0} e^{-k t}$ represents the drug concentration $A(t)$ (in $\mu \mathrm{g} / \mathrm{dL}$ ) in the bloodstream $t$ hours after injection. The value of $k$ is a constant related to the rate at which the drug is removed by the body.
a. Substitute 0.69 for $A(t)$ and 3 for $t$ in the model and write the resulting equation.
b. Substitute 0.655 for $A(t)$ and 4 for $t$ in the model and write the resulting equation.
c. Use the system of equations from parts (a) and (b) to solve for $k .$ Round to 3 decimal places.
d. Use the system of equations from parts (a) and (b) to approximate the initial concentration $A_{0}$ (in $\mu \mathrm{g} / \mathrm{dL}$ ) at the time of injection. Round to 2 decimal places.
e. Determine the concentration of the drug after $12 \mathrm{hr}$. Round to 2 decimal places.