Problem 18 - Algebra 1: Concepts and Skills

Question

In Exercises 18 and \(19,\) use the following information. For a particular fire hose, the flow rate \(f\) (in gallons per minute) can be modeled by \(f=120 \mathrm{V} p,\) where \(p\) is the nozzle pressure in pounds per square inch. Find the domain of the flow rate model. Then sketch its graph.

Step-by-Step Solution

Answer

The domain of the flow rate model is \(p \geq 0\). The graph is a straight line starting at the origin (0,0) and sloping upwards with a slope of \(120\).

Step 1: Identify Function

Identify the function given in the problem, which is \(f=120p\). This function describes the flow rate \(f\) of a fire hose in terms of the nozzle pressure \(p\).

Step 2: Determine Domain

The domain of a function is the set of all possible input values (in this case, the value of \(p\)). Since pressure cannot be negative, the domain of the function is \(p \geq 0\).

Step 3: Sketch the Graph

This function is a linear function with a slope of \(120\). Since the pressure \(p\) is greater than or equal to \(0\), the graph will start from the origin (0,0) and slope upwards, showcasing that as the pressure increases, so does the flow rate.

Key Concepts

Domain of a Function

The domain of a function represents all the possible input values that the function can accept. In our context, we are dealing with the flow rate of a fire hose, which is a function of the nozzle pressure, denoted as \( f = 120p \). The nozzle pressure, \( p \), is measured in pounds per square inch.

Understanding the domain is crucial in determining the applicability and limitations of our model. Since pressure cannot be a negative value in practical scenarios, the domain is restricted to non-negative values. Thus, the domain is all real numbers \( p \) such that \( p \geq 0 \). This is because it is physically impossible to have a negative pressure, making non-negative values the only feasible inputs.

Determining the domain correctly ensures the model is realistic and reflects actual conditions where the fire hose operates.

Graphing Linear Equations

Graphing linear equations involves visually representing the relationship between the dependent and independent variables. Here, the equation \( f = 120p \) outlines a direct and proportional relationship between the flow rate \( f \) and nozzle pressure \( p \). This linear relationship exudes simplicity yet maintains a practical implication.

The graph of a linear equation like this one is a straight line. For the equation given, the line passes through the origin (0,0), starting at the point where both pressure and flow rate are zero. As the pressure \( p \) increases, the flow rate \( f \) increases at a consistent rate—this is known as the slope of the line, which is 120. This slope indicates that for each unit increase in pressure, the flow rate increases by 120 gallons per minute.

To graph this:

• Mark the origin (0,0) as the starting point of the graph.
• Since it's a line, extend it upwards in a manner where each step right along the x-axis (pressure, \( p \)) results in a 120-unit step upwards along the y-axis (flow rate, \( f \)).

This method creates a visual representation, providing an intuitive understanding of the impact of changes in nozzle pressure on flow rate.

Modeling Real-World Scenarios

Models like \( f = 120p \) are crucial in translating real-world phenomena into mathematical terms, allowing us to predict and analyze behavior under varying conditions. This specific model is useful in emergency services, helping firefighters understand how changes in pressure affect water delivery rates, crucial for firefighting efficiency.

In real-world scenarios, mathematical modeling simplifies complex systems, making them easier to manipulate and understand.

Here's why it's beneficial:

• It assists in planning for different emergency situations by predicting the necessary pressure to achieve required flow rates.
• Helps in conducting simulations to test fire hose systems under various conditions without real-life trial, thus saving resources.
• Enables quick decision-making by providing data-backed predictions, ultimately enhancing response effectiveness in firefighting.

Mathematical modeling like this is an essential part of many industries, where predicting outcomes and optimizing performance are critical.