Problem 70 - College Algebra and Calculus: An Applied Approach

Question

Doctors treated a patient at an emergency room from 2:00 P.M. to 7:00 P.M. The patient's blood oxygen level \(L\) (in percent) during this time period can be modeled by $$ L=-0.270 t^{2}+3.59 t+83.1, \quad 2 \leq t \leq 7 $$ where \(t\) represents the time of day, with \(t=2\) corresponding to 2:00 P.M. Use the model to estimate the time (rounded to the nearest hour) when the patient's blood oxygen level was \(93 \%\). 70\. Prescription Drugs The total amounts \(A\) (in billions of dollars) projected by the industry to be spent on prescription drugs in the United States from 2002 to 2012 can be approximated by the model. $$ A=0.89 t^{2}+15.9 t+126, \quad 2 \leq t \leq 12 $$ where \(t\) represents the year, with \(t=2\) corresponding to 2002\. Use the model to predict the year in which the total amount spent on prescription drugs will be about \(\$ 374,000,000,000\). (Source: U.S. Center for Medicine and Medicaid Services)

Step-by-Step Solution

Answer

The blood oxygen level of the patient was at 93% around 3:00 P.M., while the total expenditure on prescription drugs will be about $374 billion around the year 2008.

Step 1: Problem 1, Step 1: Set up the Equation

The blood oxygen level can be modeled by the equation \(L=-0.270 t^{2}+3.59 t+83.1\). Substitute \(L\) with \(93\).

Step 2: Problem 1, Step 2: Solve the equation

Solve the quadratic equation \(-0.270 t^{2}+3.59 t+83.1 = 93\) for \(t\). This can be done using the quadratic formula \(t = \frac{-b \pm \sqrt{b^{2}-4ac}}{2a}\). If we get two valid solutions for \(t\), only consider the values that are within the range [2,7].

Step 3: Problem 1, Step 3: Round the solution

The solution \(t\) will be in terms of time past 2:00 P.M., so round it to the nearest hour.

Step 4: Problem 2, Step 1: Set up the Equation

The total amount spent on drugs can be modeled by the equation \(A=0.89 t^{2}+15.9 t+126\). Substitute \(A\) with \(374\).

Step 5: Problem 2, Step 2: Solve the Equation

Solve the quadratic equation \(0.89 t^{2}+15.9 t+126 = 374\) for \(t\). Again, apply the quadratic formula, and consider the time period constraints [2,12].

Step 6: Problem 2, Step 3: Formulate the final answer

The solution \(t\) will correspond to the year past 2002. Therefore, add 2002 to \(t\) to get the year.

Key Concepts

Blood Oxygen Level Modeling

Modeling blood oxygen levels is a critical aspect of healthcare, providing insights that can inform treatment plans. In the given exercise, a patient's oxygen saturation over time is represented by a quadratic equation, which is shaped like a parabola. This equation, \( L=-0.270t^2+3.59t+83.1 \), takes into account the time of day to predict the oxygen level in percentage.

The use of a quadratic equation in this context allows for the mapping of a non-linear relationship between time and blood oxygen levels, which is more realistic than a straight-line graph. As bodily functions such as blood oxygenation don't change at a constant rate, the parabolic model accommodates periods of rapid change and leveling off. This helps doctors to understand patterns and foresee critical changes in a patient's condition.

Prescription Drug Spending Prediction

Economic projections such as prescription drug spending can greatly benefit from the application of quadratic equations. The exercise demonstrates how a quadratic model, \( A=0.89t^2+15.9t+126 \), can be used to predict the spending on prescription drugs over a span of years. The shape of the curve produced by this equation indicates the acceleration of spending over time, not just the increase itself.

This model offers a simple yet powerful method for visualizing trends and making forecast estimates. By inputting different years into the equation, one can anticipate future spending and plan accordingly. This can aid policymakers and companies within the healthcare sector to prepare budgets, resources, and strategies to meet projected demands.

Quadratic Formula

The quadratic formula, \( t = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \), is a powerful tool in solving quadratic equations. It is universally applicable for all quadratic equations where \( ax^2 + bx + c = 0 \), providing the roots of the parabola. In both the blood oxygen level and prescription spending problems, the quadratic formula is used to find specific values of \( t \) that satisfy the given conditions.

Understanding how to manipulate and solve the quadratic formula is essential for students as it's frequently employed in various scientific and economic fields. It involves finding the discriminant, \( b^2-4ac \), which reveals the nature of the roots, and then calculating the solutions that fall within the specified domain of the problem.

Mathematical Modeling in Healthcare

Mathematical modeling is a cornerstone of healthcare innovation and management. It can help predict patient outcomes, the spread of diseases, and the economic impact of health-related policies. These models, often underpinned by equations like those for blood oxygen level and prescription drug spending, transform vast amounts of data into actionable insights.

In healthcare, accurate mathematical models can mean the difference between life and death, making comprehending and applying these tools critical for healthcare professionals. Students learning about these applications should appreciate the complexity of biological systems and the importance of precise calculations when it comes to healthcare administration and patient care.