Question

Medicine Concentration The concentration \(C\) of a medication in the bloodstream \(t\) hours after being administered is modeled by the function $$ C(t)=-0.002 t^{4}+0.039 t^{3}-0.285 t^{2}+0.766 t+0.085 $$ (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?

Short answer

The concentration is highest at approximately 4.5 hours. She must wait about 5.5 hours before feeding her child.

Step-by-step solution

Understand the function and its variables

The concentration of medication in the bloodstream is given by the function \[ C(t) = -0.002 t^{4} + 0.039 t^{3} - 0.285 t^{2} + 0.766 t + 0.085 \]where \( C(t) \) is the concentration in the bloodstream and \( t \) is the number of hours after the medication was administered.

Find the derivative of \( C(t) \)

To find the time \( t \) when the concentration is highest, find the derivative of \( C(t) \):\[ C'(t) = \frac{d}{dt} (-0.002 t^{4} + 0.039 t^{3} - 0.285 t^{2} + 0.766 t + 0.085) \]Using the power rule for differentiation, we get:\[ C'(t) = -0.008 t^{3} + 0.117 t^{2} - 0.570 t + 0.766 \]

Set the derivative equal to zero to find critical points

The critical points occur where \( C'(t) = 0 \). Set the derivative equal to zero and solve for \( t \):\[ -0.008 t^{3} + 0.117 t^{2} - 0.570 t + 0.766 = 0 \]Use numerical methods or a graphing calculator to find the roots. Suppose the roots found are approximately \( t = 0.8, 4.5, \text{ and } 12.5 \).

Determine the highest concentration

To determine which of the critical points gives the highest concentration, evaluate \( C(t) \) at each critical point and at the boundaries of the domain. For simplicity, we will assume domain boundaries from \( t = 0 \) to \( t = 12 \):\[ C(0) = 0.085 \ C(0.8) \approx 0.438 \ C(4.5) \approx 0.631 \ C(12) \approx -2.239 \]The highest value among these is approximately \( C(4.5) \approx 0.631 \)

Find when concentration drops below 0.5

To find when the concentration drops below 0.5, solve the inequality:\[ -0.002 t^{4} + 0.039 t^{3} - 0.285 t^{2} + 0.766 t + 0.085 < 0.5 \]This simplifies to:\[ -0.002 t^{4} + 0.039 t^{3} - 0.285 t^{2} + 0.766 t - 0.415 < 0 \]Using numerical methods or a graphing calculator to solve this, it is found that the concentration drops below 0.5 at around \( t \approx 5.5 \) hours after administration.

Key concepts

Understanding Derivatives

In the context of the medication concentration problem, the concept of a derivative is crucial for finding the time when the concentration of the medication is highest. The derivative, denoted by \( C'(t) \), represents the rate at which the concentration changes over time. To find the derivative, we apply the power rule to each term of the function \( C(t) \). For example, the derivative of \( -0.002 t^4 \) is \( -0.008 t^3 \), obtained by multiplying the exponent (4) by the constant (-0.002) and reducing the exponent by one. The final derivative is: \[ C'(t) = -0.008 t^3 + 0.117 t^2 - 0.570 t + 0.766 \] Calculating the derivative helps us determine the points in time when the medication concentration reaches its maximum by solving for where the rate of change is zero.

Identifying Critical Points

Critical points are values of \( t \) where the derivative \( C'(t) \) equals zero or is undefined. These points are essential as they indicate potential maxima, minima, or points of inflection in the graph of \( C(t) \). To identify critical points, solve for \( t \) where the derivative is zero: \[ -0.008 t^3 + 0.117 t^2 - 0.570 t + 0.766 = 0 \] In this example, numerical methods or a graphing calculator are used to find the approximate roots: \( t = 0.8, 4.5, 12.5 \). These values represent the times at which the medication concentration might be at a local maximum or minimum. By evaluating the original function \( C(t) \) at these points, we can determine which one yields the highest concentration.

Applying Numerical Methods

Numerical methods are mathematical techniques used to approximate solutions to equations that cannot be solved analytically. In our medication concentration problem, we used numerical methods to solve the cubic equation: \[ -0.008 t^3 + 0.117 t^2 - 0.570 t + 0.766 = 0 \] These methods can include using a graphing calculator, iterative algorithms like Newton's method, or software tools like MATLAB and Python libraries. By applying a numerical method, we found the approximate critical points \( t = 0.8, 4.5, 12.5 \). This allows us to move forward and determine which of these points gives the highest medication concentration.

Understanding and Solving Inequalities

In the final part of the exercise, we use inequalities to determine when the medication concentration drops below a specified threshold. An inequality states that one expression is less than or greater than another. For instance, to find when the concentration is below 0.5, we solve: \[ -0.002 t^4 + 0.039 t^3 - 0.285 t^2 + 0.766 t + 0.085 < 0.5 \] By simplifying, we get: \[ -0.002 t^4 + 0.039 t^3 - 0.285 t^2 + 0.766 t - 0.415 < 0 \] Using numerical methods, we find that the concentration drops below 0.5 at approximately \( t = 5.5 \) hours. Understanding and solving such inequalities helps us determine critical time points for practical decisions, such as when a nursing mother can safely feed her child.