Problem 83 - College Algebra

Question

For the following exercises, use the given rational function to answer the question. The concentration \(C\) of a drug in a patient's bloodstream \(t\) hours after injection is given by \(C(t)=\frac{100 t}{2 t^{2}+75} .\) Use a calculator to approximate the time when the concentration is highest.

Step-by-Step Solution

Answer

The concentration is highest approximately 6.12 hours after injection.

Step 1: Understanding the Function

The function given is a rational function representing the concentration of a drug at time \( t \). It is written as \( C(t) = \frac{100t}{2t^2 + 75} \). We need to find the value of \( t \) where \( C(t) \) is maximized.

Step 2: Finding Critical Points

To find the maximum concentration, we must find the critical points of the function. These occur where the first derivative, \( C'(t) \), is zero or undefined. We start by finding \( C'(t) \) using the quotient rule.

Step 3: Applying the Quotient Rule

The quotient rule is \( \frac{d}{dt} \left( \frac{u}{v} \right) = \frac{u'v - uv'}{v^2} \). For \( C(t) \), where \( u = 100t \) and \( v = 2t^2 + 75 \), find the derivatives: \( u' = 100 \) and \( v' = 4t \). Substitute to get \( C'(t) = \frac{(100)(2t^2 + 75) - (100t)(4t)}{(2t^2 + 75)^2} \).

Step 4: Simplifying the Derivative

Simplify the numerator of \( C'(t) \) to find critical points. \( 100(2t^2 + 75) \) simplifies to \( 200t^2 + 7500 \). \( 100t \cdot 4t \) is \( 400t^2 \). Therefore, the numerator becomes \( 200t^2 + 7500 - 400t^2 = -200t^2 + 7500 \).

Step 5: Setting the Derivative to Zero

Set the simplified form of \( C'(t) \) to zero to find critical points: \( -200t^2 + 7500 = 0 \). Solving for \( t^2 \) gives \( 200t^2 = 7500 \), or \( t^2 = 37.5 \). Thus, \( t = \sqrt{37.5} \) which is approximately 6.12.

Step 6: Evaluating the Critical Point

This critical point at \( t \approx 6.12 \) is where \( C(t) \) might reach a maximum. As the function is continuous and the first derivative changes sign, verify further with a graph or test intervals around this point to ensure it is a maximum.

Key Concepts

Critical Points

In calculus, critical points of a function are the values in the domain of a function where its derivative is zero or undefined. These points are important because they can help identify where a function may have local maximums, minimums, or points of inflection.

• For the function described in this exercise, we are interested in finding where the concentration of the drug in the patient's bloodstream is at a maximum. This occurs at the critical points of the concentration function.
• We found these points by setting the first derivative of the function to zero. In this case, the critical point occurs at approximately \( t \approx 6.12 \).

This analysis is crucial for understanding the behavior of rational functions and can provide insight into the effectiveness of the drug over time.

Quotient Rule

The quotient rule is a technique used in calculus to find the derivative of a fraction where both the numerator and the denominator are differentiable functions.

• The rule is given by \( \frac{d}{dt} \left( \frac{u}{v} \right) = \frac{u'v - uv'}{v^2} \), where \( u \) is the numerator function and \( v \) is the denominator function.
• For the rational function in the exercise, \( u = 100t \) and \( v = 2t^2 + 75 \), which makes their derivatives \( u' = 100 \) and \( v' = 4t \) respectively.

Using these derivatives, we were able to apply the quotient rule to find the first derivative of the function, which is essential for identifying critical points.

Maximizing Functions

Maximizing a function involves finding the point at which it achieves its highest value, given its domain.

• This is particularly useful in many practical applications, such as determining optimal dosing times in medicine.
• For a rational function like the one in this example, we apply calculus to find where the first derivative is zero, indicating potential maximums (or minimums).
• After finding critical points through differentiation, we usually confirm whether these points are maximums by analyzing the function's behavior around these points.

Here, we found that the maximum concentration occurs at around 6.12 hours after the injection, which is crucial for effectively managing a patient's medication regime.

First Derivative

The first derivative of a function provides valuable information about the function's rate of change. It can indicate how a function is increasing or decreasing at any point along its domain.

• By taking the first derivative, you determine where the slopes of tangent lines are zero, pinpointing the critical points where the function might have local extremes.
• In our scenario with the concentration function \( C(t) = \frac{100t}{2t^2 + 75} \), the first derivative was essential to find where the concentration changes direction, highlighting potential maximum or minimum concentration levels.

Understanding the first derivative is a cornerstone of analyzing rational functions and is crucial for solving many real-world calculus problems.