Problem 19 - 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. If the nozzle pressure is 100 pounds per square inch, what is the flow rate?

Step-by-Step Solution

Answer

The exercise is missing essential information (the value of \(V\)). Therefore, it is not possible to calculate the flow rate.

Step 1: Analyze the Given Information

From the problem, it's known that the flow rate \(f\) can be calculated with the formula \( f = 120 V p \), and the value of \(p\) is given as 100 pounds per square inch. But the value of \(V\) is not known.

Step 2: Identify the Missing Information

The unknown from the provided information is the variable \(V\). Without it, the formula cannot be solved. Therefore, the task is missing essential information to solve the problem.

Step 3: Inform about the Missing Information

In this case, it is not possible to solve the task without the missing value for \(V\). This needs to be communicated.

Key Concepts

Nozzle Pressure

Nozzle pressure is a fundamental concept in understanding how fluid dynamics work, especially in the context of fire fighting. It refers to the pressure exerted by the fluid as it exits the hose through the nozzle. At a nozzle, pressure is crucial as it determines the force with which water is expelled, impacting the flow rate and the distance the water can reach.
In fire fighting, nozzle pressure is measured in pounds per square inch (psi). When the nozzle pressure is higher, more forceful streams of water are produced, which can be essential for reaching high or distant flames. Accurate measuring and control of nozzle pressure ensure effective disaster management and resource allocation, allowing firefighters to efficiently manage water resources during an emergency.
Keeping in mind the formula given in the exercise — where the flow rate is impacted by nozzle pressure — students can see how critical this component is to fire hose operation.

Gallons per Minute

Gallons per minute (GPM) is the standard measurement of flow rate, particularly in contexts like fire suppression. It specifies how many gallons of water flow from a hose or pipe every minute. For firefighters, knowing the GPM is crucial as it indicates how much water is being used and whether it will be sufficient to extinguish a fire.
The exercise presented uses the formula \(f = 120 V p\), where \(f\) represents the flow rate measured in GPM, \(p\) signifies the nozzle pressure, and \(V\) is an unknown variable that acts as a proportional constant. This setup underlines the importance of GPM as it associates flow rate with actual firefighting scenarios. Practically, GPM helps in deciding which nozzles to use, as different fires might require different flow rates to be controlled effectively.
Understanding GPM allows students to gain insights into fluid mechanics and reinforces the intricacies of managing real-world fire emergencies.

Fire Hose Flow Model

The fire hose flow model serves as a critical application of fluid dynamics by linking pressure, flow, and volume. In the given problem, the flow model is represented by the equation \(f = 120 V p\). This formula helps to quantify how water is transported through a hose and expelled from the nozzle, providing insight into the mechanics that firefighters rely on.
This model uses key variables:

• \(f\) — the flow rate in gallons per minute, describing the volume of water expelled.
• \(p\) — nozzle pressure in psi, which influences how strongly the water is projected from the nozzle.
• \(V\) — a constant or factor that ties flow and pressure together, though the exercise did not provide its value.

Fire hose flow models are essential for precise water deployment, which directly impacts the success of extinguishing fires. By understanding the flow model, students can appreciate how adjusting the factors leads to different outcomes on the field, crucial for both theoretical and practical firefighting knowledge.