Problem 104 - College Algebra and Calculus: An Applied Approach

Question

The average price \(G\) (in dollars) of generic prescription drugs from 1998 to 2005 can be modeled by \(G=2.005 t+0.40, \quad 8 \leq t \leq 15\) where \(t\) represents the year, with \(t=8\) corresponding to \(1998 .\) Use the model to find the year in which the price of the average generic drug prescription exceeded \(\$ 19\).

Step-by-Step Solution

Answer

The average price of generic prescription drugs exceeded \$19 in 2007.

Step 1: Set up the equation

First, set up the equation by setting \(G\) to 19 as we need to find the year when the price exceeded this number: \[19 = 2.005t + 0.40.\]

Step 2: Solve for \(t\)

Next, solve for \(t\). Start by subtracting 0.40 from both sides of the equation, which gives us \[18.6 = 2.005t.\] Then, divide both sides by 2.005 to isolate \(t\), which results in \(t \approx 9.27\) after rounding to two decimal places.

Step 3: Interpret the result

The result means that the average price of generic prescription drugs exceeded \$19 in the year represented by \(t \approx 9.27\). Adding 9 to 1998 gives us the year 2007. The fractional part indicates that the price exceeded \$19 at some point after 2007 began, but we can't determine the exact date from this model. Therefore, we can conclude that the price exceeded \$19 during 2007.

Key Concepts

Linear Equations

Linear equations form the backbone of algebraic modeling, allowing us to understand relationships between variables. In our exercise, the equation \( G = 2.005t + 0.40 \) represents a linear relationship between the year \( t \) and the average price of generic prescription drugs \( G \).

Linear equations are characterized by having one or more variables that increase consistently as other factors increase. They appear in the form \( y = mx + c \), where \( m \) is the slope, and \( c \) is the y-intercept. The slope illustrates how much \( y \) changes for each unit increase in \( x \). In this case:

• \( m = 2.005 \): Reflects the rate at which the price increases per year.
• \( c = 0.40 \): Represents the base price value of the drug in our model during the initial year.

This equation helps us predict future costs if the current rate continues unchanged. Knowing the linear structure allows easy manipulation of equations to find unknowns, such as which year corresponds to a specific price. Thus, through the use of linear equations, algebraic modeling becomes a powerful tool.

Year Calculation

Calculating years based on given models involves understanding variables and correctly interpreting what they represent. In the given equation \( G = 2.005t + 0.40 \), the parameter \( t \) refers to years, but it's crucial to remember how \( t \) is defined in context.

In our exercise, \( t = 8 \) corresponds to the year 1998. Therefore, \( t \) indicates the number of years past 1990. Once we solve for \( t \), it's vital to convert this into a calendar year:

• We found \( t \approx 9.27 \) as the point where \( G = 19 \).
• Add this value to 1998. Since \( t \approx 9.27 \), the beginning of 2007 corresponds to when the price first exceeded \$19.

It's important to note that \( 0.27 \) indicates the fraction of the year, suggesting the transition occurs sometime into the year. While specific dates aren't provided, this emphasizes our understanding that algebraic modeling often gives a broader, rather than a pinpoint, timeframe. Thoroughly understanding this trend ensures we interpret results meaningfully.

Data Interpretation

Data interpretation involves deriving meaningful insights from numerical models like our linear equation for drug prices. Here, our focus is on using the model to predict when an event occurs—in this case, when generic drug prices exceed a certain value.

The given model provides an efficient way to interpret data over time because:

• It maps data points (years) to outcomes (drug prices).
• Understanding this map helps estimate the year when conditions change, like surpassing the price threshold of \\(19.

In data interpretation, determining the relevance of your findings is crucial. In our scenario, knowing that prices crossed \\)19 in 2007 could highlight shifts in market trends or healthcare policies.

Successful interpretation also involves addressing what the model doesn't tell us—like specific months or external factors affecting price changes. This highlights the importance of considering limitations and seeking additional data to form a comprehensive view. Interpretations thus not only provide answers but also refine our questions, driving a deeper understanding of economic conditions and potentially influencing future actions.