Problem 33 - College Algebra with Modeling and Visualization

Question

The function given by \(P(x)=19.4 x\) calculates the pounds of \(\mathrm{CO}_{2}\) (carbon dioxide) released into the atmosphere by a car burning \(x\) gallons of gasoline. (a) Calculate \(P(20)\) and interpret the result. (b) Find the slope of the graph of \(P\). Interpret this slope as a rate of change.

Step-by-Step Solution

Answer

(a) 388 pounds of \(\mathrm{CO}_2\). (b) 19.4 pounds of \(\mathrm{CO}_2\) per gallon.

Step 1: Calculate P(20)

To find \(P(20)\), substitute \(x = 20\) into the function \(P(x) = 19.4x\). This gives us \(P(20) = 19.4 \times 20\). Calculating this, we get \(P(20) = 388\). So, burning 20 gallons of gasoline releases 388 pounds of \(\mathrm{CO}_2\) into the atmosphere.

Step 2: Calculate the Slope

The function \(P(x) = 19.4x\) is in the form of \(P(x) = mx + b\), where \(m\) is the slope. Here, \(m = 19.4\). This means the slope of the graph of \(P\) is 19.4, which represents the rate of change.

Step 3: Interpret the Slope

The slope of 19.4 indicates that with each additional gallon of gasoline burned, 19.4 pounds of \(\mathrm{CO}_2\) are released into the atmosphere. This quantifies the amount of \(\mathrm{CO}_2\) emissions per gallon of gasoline.

Key Concepts

CO2 Emissions

Understanding CO2 emissions is important when discussing the environmental impact of burning gasoline. Cars emit carbon dioxide (\(\mathrm{CO}_2\)) when gasoline is burnt. In our example, the function \(P(x) = 19.4x\) helps quantify this.

Each value of \(x\) in the function represents the gallons of gasoline burnt, and \(P(x)\) gives the resulting \(\mathrm{CO}_2\) emissions in pounds. This direct relationship between gasoline burnt and \(\mathrm{CO}_2\) produced makes it straightforward to calculate emissions for different amounts of gasoline used.

For instance, burning 20 gallons of gasoline emits 388 pounds of \(\mathrm{CO}_2\) (calculated by \(P(20)\)). This reveals how much \(\mathrm{CO}_2\) is released directly due to fuel consumption and highlights the environmental footprint left by vehicles.

Rate of Change

The concept of rate of change is crucial in understanding how one variable impacts another, especially in linear functions. In the given function \(P(x) = 19.4x\), the rate of change is represented by the coefficient 19.4.

• The rate of change tells us the incremental increase in \(\mathrm{CO}_2\) emissions with each gallon of gasoline burned.
• Here, for each additional gallon, 19.4 pounds of \(\mathrm{CO}_2\) are added to the atmosphere.

This measure gives insight into how significant even small changes in gasoline consumption can be on a larger scale, emphasizing the need to monitor and regulate fuel efficiency in vehicles.

Slope Interpretation

The slope in a linear function like \(P(x) = mx + b\) provides valuable insights into the relationship between the variables involved. In the equation \(P(x) = 19.4x\), the slope \(m = 19.4\) is crucial for interpreting this relationship.

• The slope tells us that each unit increase in \(x\) (each extra gallon of gasoline) results in a direct increase of 19.4 in \(P(x)\) (pounds of \(\mathrm{CO}_2\)).
• This quantifies the constant relationship between gasoline consumption and \(\mathrm{CO}_2\) emissions.

Understanding the slope helps us predict the impact of gasoline consumption changes on emissions, enabling more informed decisions on energy use and policy-making for environmental protection.