The concentration of a drug in the body decreases exponentially after a dosage is given. In one clinical study, adult subjects averaged $14 \mathrm{micrograms} / \mathrm{milliliter}(\mathrm{mcg} / \mathrm{mL}$ ) of the drug in their blood plasma $1 \mathrm{hr}$ after a 1000-mg dosage and 3 micrograms/milliliter $6 \mathrm{hr}$ after dosage. Assume the concentration decreases according to the exponential decay model. a) Find the value $k$, and write an equation for an exponential function that can be used to predict the concentration of the drug, in micrograms/milliliter, $t$ hours after a 1000 -mg dosage. b) Estimate the concentration of the drug $2 \mathrm{hr}$ after a $1000-\mathrm{mg}$ dosage. c) To relieve a fever, the concentration of the drug should go no lower than $5 \mathrm{mcg} / \mathrm{mL}$. After how many hours will a 1000 -mg dosage drop to that level? a) $k=\square$ (Round to three decimal places as needed.)

Step 1 ：The general form of an exponential decay function is \(y = a \cdot e^{kt}\), where \(a\) is the initial amount, \(k\) is the decay constant, and \(t\) is time.

Step 2 ：In this case, we know that \(a = 14\) (the initial concentration of the drug), and we have two points on the decay curve: \((1, 14)\) and \((6, 3)\). We can use these two points to form two equations and solve for \(k\).

Step 3 ：First, we can write the equation for the decay model as \(y = 14 \cdot e^{kt}\).

Step 4 ：Then, we can substitute the two points into the equation to get two equations: 1) \(14 = 14 \cdot e^{k}\) 2) \(3 = 14 \cdot e^{6k}\)

Step 5 ：We can solve these two equations to find the value of \(k\). The first equation gives us \(k = 0\), which is not possible in an exponential decay model. The second equation gives us multiple solutions, some of which are complex numbers. We are looking for a real number solution.

Step 6 ：We can see that the real part of the solution is \(log(14^{5/6}*3^{1/6}/14)\), which is the solution we are looking for.

Step 7 ：Let's calculate this value. The value of \(k\) is \(\boxed{-0.257}\) (rounded to three decimal places).