Chapter 8.4, Problem 59PE

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 $3\text{ hr}$ , the drug concentration in the blood is $0.69\mu g\text{/}dL$ and after $4\text{ hr,}$ the concentration is $0.655\mu g\text{/dL}.$ The model $A\left(t\right)=A_0{e}^{-kt}$ represents the drug concentration $A\left(t\right)\left(\text{in }\mu g\text{/dL}\right)$ 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 $A\left(\text{t}\right)$ and $3$ for $t$ in the model and write the resulting equation.

b. Substitute $0.655$ for $A\left(\text{t}\right)$ 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 $A_0$ $\left(\text{in }\mu g\text{/dL}\right)$ at the time of injection. Round to $2$ decimal places.

e. Determine the concentration of the drug after $12\text{ hr}\text{.}$ Round to $2$ decimal places.