Q 76. - Precalculus Enhanced with Graphing Utilities

Question

Medicine Concentration: The concentration C of a medication in the bloodstream t hours after being administered is modeled by the function $C\left(t\right)=-0.002t4+0.039t3-0.285t2+0.766t+0.085$.

(a) After how many hours will the concentration be highest?

(b) A woman nursing a child must wait until the concentration is below $0.5$ before she can feed him. After taking the medication, how long must she wait before feeding her child?

Step-by-Step Solution

Answer

Part (a). The graph of the given function is shown below:

The concentration is the highest at $2\text{hr}$.

Part (b). The graph of the given function at the concentarion  is shown below:

The the woman should wait for $4.465\text{hr}$ after taking the medicine.

Part (a). Step 1. Given information

The given function is $C\left(t\right)=-0.002t^4+0.039t^3-0.285t^2+0.766t+0.085$.

Part (a) Step 2. Graph the given function by using graphic utility.

The graph of the given function is shown:

Part (a) Step 3. Find the number of hours when the concentration is highest.

From the graph, observe that the concentarition is highent when the time is $2.16\text{hr}\approx 2\text{hr}$

Part (b) Step 1. Draw the graph of the given function to determine the time when the concentartion is 0 . 5 .

The graph is shown below:

Part (b) Step 2. Determine the time when the concentration drops below 0 . 5 in order to feed the child.

From the graph, observe that the concentraion drops below $0.5$ after the time $4.465\text{hr}$.

Hence, the woman should wait for $4.465\text{hr}$ after taking the medicine.