Problem 82 - Algebra and Trigonometry with Analytic Geometry

Question

To treat arrhythmia (irregular heartbeat), a drug is fed intravenously into the bloodstream. Suppose that the concentration \(c\) of the drug after \(t\) hours is given by \(c=3.5 t /(t+1) \mathrm{mg} / \mathrm{L}\). If the minimum therapeutic level is \(1.5 \mathrm{mg} / \mathrm{L}\), determine when this level is exceeded.

Step-by-Step Solution

Answer

The concentration exceeds the therapeutic level after 0.75 hours.

Step 1: Set up the equation

To find out when the concentration exceeds the minimum therapeutic level of \(1.5 \text{ mg/L}\), we set the concentration equation \(c = \frac{3.5t}{t+1}\) equal to \(1.5\). This gives us the inequality \(\frac{3.5t}{t+1} > 1.5\).

Step 2: Eliminate the fraction

Multiply both sides of the inequality by \((t+1)\) to eliminate the denominator: \(3.5t > 1.5(t+1)\).

Step 3: Expand and simplify the inequality

Distribute \(1.5\) on the right side: \(3.5t > 1.5t + 1.5\). Then, rearrange the terms: \(3.5t - 1.5t > 1.5\). Simplify to get \(2t > 1.5\).

Step 4: Solve for \(t\)

Divide both sides of the inequality by \(2\) to isolate \(t\): \(t > 0.75\).

Step 5: Interpret the solution

The inequality \(t > 0.75\) means the concentration of the drug exceeds \(1.5 \text{ mg/L}\) after \(0.75\) hours.

Key Concepts

Inequality Solving

Understanding inequality solving is essential in determining when a condition is either met or exceeded. An inequality compares two values or expressions that might not be equal to each other. Instead, it shows the relative size or difference between two expressions.In the exercise provided, the inequality \(\frac{3.5t}{t+1} > 1.5\) shows that we're looking for when the concentration of the drug is greater than 1.5 mg/L. To simplify solving inequalities:

• Remove fractions by multiplying through by a common denominator.
• Rearrange terms to isolate the variable on one side.
• Be mindful of flipping the inequality sign when multiplying or dividing by negative numbers, which wasn't needed here but is key in other problems.

After simplifying the given inequality, you find \(t > 0.75\), meaning the concentration surpasses the therapeutic level after 0.75 hours.

Rational Equations

Rational equations involve fractions where the numerator and/or denominator contains a variable. In our situation, the concentration \(c=\frac{3.5t}{t+1}\) is a rational expression because it involves division by \((t+1)\).To solve rational equations:

• Identify any restrictions on the variable, often resulting from situations where the denominator can’t be zero.
• Clear the fractions by multiplying through by the common denominator, which was done when transitioning to inequality \(3.5t > 1.5(t+1)\).
• After clearing the fractions, solve like any linear equation or inequality, by isolating the variable.

Being familiar with these steps allows for solving more intricate rational equations you might encounter in other exercises.

Concentration Problems

Concentration problems like this one often appear in real-world applications such as medicine and chemistry. These problems frequently involve understanding how the concentration of a substance changes over time.When solving a concentration problem, you often deal with:

• A function describing the concentration over time, as seen with \(c = \frac{3.5t}{t+1}\).
• Determining critical points where concentration levels meet certain conditions, such as therapeutic minimums or toxic maximums.
• Using algebraic techniques, like setting inequalities, to find these times.

In this problem, you found that the concentration level exceeded 1.5 mg/L after 0.75 hours, providing crucial information on dosing and patient safety.