Chapter 2.3, Problem 76AYU

80. Medicine Concentration The concentration C of a medication in the bloodstream t hours after being administered is modeled by the function

$C\left(t\right) = -O.OOZ{x}^4 + 0.039f^3 - 0.285i^2 + 0.766i + 0.085$

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

(b) (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?

To determine

To find: The following values using graphing utility,

a. Graph the function $C=C\left(t\right)=-0.002t^4+0.039t^3-0.285t^2+0.766t+0.085$.

Answer to Problem 76AYU

a.

Explanation of Solution

Given:

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

Calculation:

It is given that the function is defined as $C=C\left(t\right)=-0.002t^4+0.039t^3-0.285t^2+0.766t+0.085$, where the concentration, $C$ of a medication in the bloodstream $t$ hours after being administered.

a.

To determine

To find: The following values using graphing utility,

b. Time to reach highest concentration.

Answer to Problem 76AYU

b. 2 hours, 9 minutes and 36 seconds to reach the highest concentration.

Explanation of Solution

Given:

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

Calculation:

It is given that the function is defined as $C=C\left(t\right)=-0.002t^4+0.039t^3-0.285t^2+0.766t+0.085$, where the concentration, $C$ of a medication in the bloodstream $t$ hours after being administered.

b. From the graph, the maximum concentration is at extremum point $x\text{-axis}$ gives the time in hours and $y\text{-axis}$ denotes the concentration. The maximum time taken is $2.16$ concentration.

To determine

To find: The following values using graphing utility,

c. Time to wait a woman before feeding her child if the concentration is below $0.5$ before she can feed him.

Answer to Problem 76AYU

c. The woman needs to wait 24 minutes before feeding her child.

Explanation of Solution

Given:

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

Calculation:

It is given that the function is defined as $C=C\left(t\right)=-0.002t^4+0.039t^3-0.285t^2+0.766t+0.085$, where the concentration, $C$ of a medication in the bloodstream $t$ hours after being administered.

c. Time to wait a woman before feeding her child if the concentration is below $0.5$ before she can feed him.

From the graph, it can be concluded that the below $0.5$ hours goes to the corresponding concentration $0.4$. That is, the woman needs to wait 24 minutes before feeding her child.