Problem 18 - Algebra 1

Question

For a fire hose with a nozzle that has a diameter of 2 inches, the flow rate \(f\) (in gallons per minute) can be modeled by \(f=120 \sqrt{p}\) where \(p\) is the nozzle pressure in pounds per square inch. Sketch a graph of the model.

Step-by-Step Solution

Answer

The graph of the model is a square root curve that starts from the origin and rises. It represents that the flow rate increases with the square root of the pressure.

Step 1: Interpret the problem

The relationship between \(f\) and \(p\) is described by the equation \(f=120 \sqrt{p}\). This equation represents a square root function. It indicates that the flow rate (\(f\)) is directly proportional to the square root of the nozzle pressure (\(p\)). A square root function graph commonly starts from the origin and rises. In this case, however, it will only rise since pressure cannot be negative.

Step 2: Determine the points

Before sketching, it's important to determine a set of points based on the function that will be used for the sketch. A table of values will help. Start with a pressure of 0 PSI: \[f=120 \sqrt{0} = 0\], so the first point is (0,0). Now, choose a pressure value, let's say 1 PSI: \[f=120 \sqrt{1} = 120\], so the second point is (1,120). To provide a more accurate sketch, you can add more relevant points.

Step 3: Sketch the graph

With the points, sketch the square root function on a Cartesian plane, taking into account that the flow rate cannot be negative and it starts from the origin. Label the x-axis as Pressure (p) and the y-axis as Flow Rate (f). The points plotted should give a slight curve that rises quickly at first then flattens out, representing the square root curve derived from the given model. The curve should start from the origin (0,0) moving through the point (1,120) and continuing to rise, albeit at a slower rate as pressure increases.

Key Concepts

Flow Rate

Flow rate is an essential concept in understanding how liquids move through different systems, like hoses and pipes. In this exercise, the flow rate (\( f \)) is the amount of water moving through the fire hose, measured in gallons per minute (GPM). The equation \( f = 120 \sqrt{p} \) tells us how the flow rate is affected by the nozzle pressure (\( p \)).

The example uses a fire hose with a nozzle diameter of 2 inches to illustrate this relationship. Knowing how various factors, like pressure, affect flow rate can help in many practical applications, including firefighting and irrigation.

In this case, the equation represents a square root function, indicating that as the pressure increases, the flow rate also increases. However, it does not increase at the same speed throughout. At first, a small increase in pressure leads to a larger increase in flow rate. As pressure continues to rise, the rate of increase in flow rate gradually slows. This is due to the nature of the square root function.

Understanding the flow rate allows for effective planning and control of how water or any liquid should be supplied through a system. This knowledge helps ensure that the correct amount of liquid is delivered for desired operations.

Nozzle Pressure

Nozzle pressure (\( p \)) is a critical factor that affects the flow rate of water through a hose. It refers to the pressure exerted by the liquid when it passes through the nozzle. It is typically measured in pounds per square inch (PSI).

In the equation provided, the pressure directly influences the flow rate through a square root relationship. As the pressure increases, the flow rate of water expelled through the nozzle rises as well.

Just like flow rate, nozzle pressure plays a vital role in various industries. For example, in firefighting, having the right pressure can make a significant difference in how effectively water is used to combat fires.

• Low pressure may result in insufficient water flow, leading to ineffective firefighting efforts.
• High pressure, on the other hand, may deliver water swiftly and in sufficient amounts to the target area.

Understanding nozzle pressure helps in managing how fluids are transported, especially in systems requiring precise delivery of liquids under varying conditions.

Graph Sketching

Creating an accurate graph based on mathematical models is an important skill. In this exercise, the graph of the function \( f = 120 \sqrt{p} \) provides a visual understanding of the relationship between pressure and flow rate.

Start by plotting points derived from the equation. At zero pressure, the flow rate is zero (\( (0,0) \)), and at 1 psi, the flow rate is 120 gallons per minute (\( (1,120) \)). By plotting these points, you begin to shape the square root curve on the Cartesian plane.

Here are some tips for graph sketching:

• Establish a scale for both the x-axis (Pressure) and y-axis (Flow Rate) to ensure all points fit within the visible part of the graph.
• Mark all significant points such as (0,0) and (1,120) first to lay a foundation for the curve.
• Understand that the curve will start steep, then become flatter as the pressure continues to increase.
• Label your axes clearly to make it easy for anyone viewing your graph to understand what it displays.

Graph sketching helps visualize complex functions and makes it easier to see how variables interact in real-world scenarios.