Problem 77 - College Algebra and Calculus: An Applied Approach

Question

Prescriptions The numbers of prescriptions \(P\) (in millions) filled in the United States from 1998 to 2005 can be modeled by \(P=-11,415+\frac{15,044}{1+e^{-0.2166 t-0.7667}}, \quad 8 \leq t \leq 15\) where \(t\) represents the year, with \(t=8\) corresponding to 1998\. (Source: National Association of Chain Drug Stores) (a) Use a graphing utility to graph \(P\) for the years 1998 to \(2005 .\) (b) Use the graph from part (a) to estimate the numbers of prescriptions filled in 1999,2002 , and 2005 .

Step-by-Step Solution

Answer

Using a graphing utility to graph the function and estimate the number of prescriptions, one can find that for the years 1999, 2002, and 2005, the approximate values for \(P(t)\) on the graph yield the estimated numbers of prescriptions for these respective years.

Step 1: Insert the Values

First, translate the years into the variable \(t\) in the given function \(P(t)\). The year 1998 corresponds to \(t=8\), so the year 1999 corresponds to \(t=9\), 2002 corresponds to \(t=12\), and 2005 corresponds to \(t=15\). Remember to replace \(t\) with these values in the function.

Step 2: Use Your Graphing Utility

Graph \(P(t)=-11,415+\frac{15,044}{1+e^{-0.2166 t-0.7667}}\) from \(t=8\) to \(t=15\) using your graphing utility. Observe where the graph is on the y-axis (millions of prescriptions) for \(t=9, t=12,\) and \(t=15\).

Step 3: Estimate the Number of Prescriptions

Estimate the number of prescriptions in 1999, 2002, and 2005 by finding the value of \(P(t)\) for \(t=9\), \(t=12\), and \(t=15\) from the graph plotted in step 2. Observe the values of \(P(t)\) at these points.

Key Concepts

Exponential Functions

Exponential functions are mathematical models often used to represent growth or decay processes. They have the form \(f(x) = ab^{x}\), where \(a\) is the initial amount, \(b\) is the base representing the rate of growth or decay, and \(x\) is the exponent. In the context of modeling prescriptions, the exponential function helps in illustrating how the number of prescriptions has changed over time.

Exponential functions are powerful because they can model complex growth rates that do not remain constant. Instead of adding a set amount each period, the growth is multiplied by a constant rate. This is because real-world scenarios like population growth or, in our case, the number of prescriptions, often do not increase linearly over time.

In the given model for prescriptions, the function includes an exponential term \(e^{-0.2166 t - 0.7667}\), which helps capture the non-linear increase in the number of prescriptions over the years.

• Exponential functions reflect real-life scenarios that involve continuous growth/decay.
• Helps predict future values based on current trends.
• Used extensively in fields such as finance, biology, and medicine.

Prescription Modeling

Prescription modeling involves creating a mathematical representation to predict or understand how prescriptions are filled over time. This is particularly useful in healthcare planning and management. In our exercise, the number of prescriptions filled from 1998 to 2005 is modeled using a function that incorporates an exponential term, making it suitable for capturing the trend of increasing prescription fills.

Using prescription models, stakeholders can:

• Anticipate future healthcare needs and allocate resources accordingly.
• Identify potential issues early, such as sudden spikes in demand.
• Understand underlying factors driving prescription trends.

Prescription modeling can provide insights into how socioeconomic factors and policies affect prescription trends, ensuring that the healthcare system is responsive to changes in demand.

Graphing Utility

A graphing utility is a tool used to plot mathematical functions accurately. This can be a software application or a physical graphing calculator. For modeling prescriptions, a graphing utility helps visualize the function \(P(t)\) and interpret its behavior over the designated time period.

Graphing utilities offer several benefits:

• Provide a visual representation of data, making trends easier to understand.
• Allow interactive exploration of different parts of the graph.
• Enable precise estimation of function values at specific points, which is crucial in our prescription exercise for years 1999, 2002, and 2005.

By inputting the given function into a graphing utility, students can observe how the number of prescriptions changes over time, highlighting peaks and allowing for better analysis.

Function Evaluation

Function evaluation is the process of calculating the output of a function based on specific input values. In the prescription modeling exercise, we evaluate the function \(P(t)\) for specific years after translating them into appropriate \(t\) values, such as \(t=9\) for 1999.

To evaluate the function, follow these steps:

• Identify the year and translate it into the variable \(t\).
• Insert this \(t\) value into the function \(P(t)\) to compute the number of prescriptions.
• Use the graphing utility's output to cross-check and approximate values if needed for validation.

This process helps verify and estimate the number of prescriptions filled in specific years, ensuring accurate forecasting and analysis for healthcare planners. Understanding function evaluation is crucial for deciphering predictions and making informed data-driven decisions.