Problem 60 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

CONCENTRATION OF A DRUG IN THE BLOODSTREAM The concentration (in milligrams/cubic centimeter) of a certain drug in a patient's bloodstream \(t\) hr after injection is given by $$ \frac{0.2 t}{t^{2}+1} $$ Find the interval of time when the concentration of the drug is greater than or equal to \(0.08 \mathrm{mg} / \mathrm{cc}\).

Step-by-Step Solution

Answer

The concentration of the drug in the patient's bloodstream is greater than or equal to \(0.08 \mathrm{mg} / \mathrm{cc}\) between 0.5 hours and 1.5 hours after injection. In other words, the desired concentration occurs in the interval \(0.5 \leq t \leq 1.5\).

Step 1: Setting up the inequality

Given the concentration function, we want to find the interval of time when the concentration is greater than or equal to 0.08 mg/cc. So, we set up the inequality: \[ \frac{0.2t}{t^2 + 1} \geq 0.08 \]

Step 2: Simplify the inequality

To solve for \(t\), we first simplify the inequality by multiplying both sides by \((t^2 + 1)\) as follows: \[ 0.2t \geq 0.08 (t^2 + 1) \] Next, we need to get the terms on one side by subtracting \(0.08(t^2 + 1)\) from both sides: \[ 0.2t - 0.08 (t^2 + 1) \geq 0 \] Now, we can distribute the \(-0.08\) to the terms inside the parentheses: \[ 0.2t - 0.08t^2 - 0.08 \geq 0 \]

Step 3: Rearrange the inequality

Rearrange the inequality to form a quadratic inequality: \[ 0.08t^2 - 0.2t + 0.08 \leq 0 \] Note: We have flipped the inequality sign because we are going to multiply by \(-1\) to make the leading coefficient positive.

Step 4: Find the critical points of the inequality

To find the critical points, we should first set the inequality as an equation and solve for \(t\): \[ 0.08t^2 - 0.2t + 0.08 = 0 \] Since this is a quadratic equation, we can use the quadratic formula to solve for \(t\): \[ t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] In our case, \(a = 0.08\), \(b = -0.2\), and \(c = 0.08\). Plugging these values into the formula, we get: \[ t = \frac{0.2 \pm \sqrt{(-0.2)^2 - 4(0.08)(0.08)}}{2(0.08)} \] Simplify the expression within the square root: \[ t = \frac{0.2 \pm \sqrt{0.04 - 0.0256}}{0.16} \] Now, calculate the square root and simplify the expression: \[ t = \frac{0.2 \pm \sqrt{0.0144}}{0.16} \] We obtain two values for \(t\): \[ t_1 = \frac{0.2 + \sqrt{0.0144}}{0.16} \quad \text{and} \quad t_2 = \frac{0.2 - \sqrt{0.0144}}{0.16} \] Calculating these, we get: \[ t_1 \approx 1.5 \quad \text{and} \quad t_2 \approx 0.5 \]

Step 5: Determine the time interval

Since we want to find when the concentration is above the desired level, we should choose the interval between the obtained critical points. Thus, the interval of time when the concentration is greater than or equal to 0.08 mg/cc is: \[ 0.5 \leq t \leq 1.5 \] So, the concentration of the drug is greater than or equal to \(0.08 \mathrm{mg} / \mathrm{cc}\) in the bloodstream between 0.5 hours and 1.5 hours after injection.

Key Concepts

Bloodstream Drug Concentration

Understanding the concentration of a drug in the bloodstream is crucial for effective medication management. The concentration tells us how much of the active ingredient, in this case, measured in milligrams per cubic centimeter (mg/cc), is present in the blood at any given time. For a medication to be effective, its concentration needs to reach a certain level without becoming toxic. In this exercise, we look at the equation that models the concentration of a drug after being administered. The formula \( \frac{0.2 t}{t^2 + 1} \) allows us to determine the concentration based on the time \(t\) hours after injection. A key aspect of solving problems related to drug concentration is identifying the interval during which its level is adequate to have the desired therapeutic effect. The equation's structure considers both the rate at which the drug enters the bloodstream and how it is metabolized over time. By examining this, we can better predict how long the effects of the drug will last.

Inequality Solving Techniques

Inequality solving techniques are essential tools in mathematics, especially when determining intervals for conditions like drug concentration levels. When you set up the inequality \( \frac{0.2t}{t^2 + 1} \geq 0.08 \), you're looking to find when the drug concentration remains above a specific threshold. This process involves several steps:

• Simplification: First, clear the fraction by multiplying both sides by the denominator, \(t^2 + 1\), to simplify the expression.
• Rearrangement: Move all terms to one side of the inequality to form a standard quadratic inequality.
• Solution: Use methods like factoring or the quadratic formula to find critical points that determine the boundary of the solution.

Reversing the inequality sign when multiplying by a negative number is crucial. This process ensures you have the correct time intervals where the concentration requirements are met. Employing these techniques helps uncover the precise time frame where conditions like drug efficacy are optimal.

Quadratic Formula Applications

The quadratic formula is a vital mathematical tool used for solving quadratic equations, which frequently arise in various scientific and engineering applications. In the context of this exercise, the formula is applied to the inequality \(0.08t^2 - 0.2t + 0.08 = 0\) to find the critical time points for a drug's concentration in the bloodstream.The formula \[ t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] provides solutions to any quadratic equation \(ax^2 + bx + c = 0\). Here:

• The coefficients \(a = 0.08\), \(b = -0.2\), and \(c = 0.08\) are plugged into the formula.
• Calculate the discriminant \( b^2 - 4ac \), which determines the number and type of solutions.
• Find the roots, which in this scenario are \(t_1\) and \(t_2\).

These roots represent the time intervals during which the quadratic inequality is satisfied and ensure the concentration stays within therapeutic limits. Knowing how to apply the quadratic formula correctly is critical for analyzing systems where changes over time need precise evaluation.