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Chapter 3: Problem 14

Horizontal reach of straight streams: If a fire hose is held horizontally, then the distance the stream will travel depends on the water pressure and on the horizontal factor for the nozzle. The horizontal factor \(H\) depends on the diameter of the nozzle. For a \(0.5\)-inch nozzle, the horizontal factor is 56 . For each \(\frac{1}{8}\)-inch increase in nozzle diameter, the horizontal factor increases by 6 . a. Explain why the function giving the horizontal factor \(H\) in terms of the nozzle diameter \(d\) (measured in inches) is linear. b. Use a formula to express \(H\) as a linear function of \(d\). c. Once the horizontal factor \(H\) is known, we can calculate the distance \(S\) in feet that a horizontal stream of water can travel by using $$ S=\sqrt{H p} \text {. } $$ Here \(p\) is pressure in pounds per square inch. How far will a horizontal stream travel if the pressure is 50 pounds per square inch and the nozzle diameter is \(1.75\) inches? d. Firefighters have a nozzle with a diameter of \(1.25\) inches. The pumper generates a pressure of 70 pounds per square inch. The hose nozzle is 75 feet from a fire. Can a horizontal stream of water reach the fire?

Short Answer

Expert verified
The function is linear because of the constant rate of change; \(H(d) = 56 + 48(d - 0.5)\); the stream travels approximately 76.16 feet for 1.75-inch nozzle; for a 1.25-inch nozzle, the stream reaches 80.25 feet, so it can reach the fire.

Step by step solution

01

Determine Linearity of Horizontal Factor

The horizontal factor increases by 6 for each \( \frac{1}{8} \)-inch increase in nozzle diameter. This constant rate of change indicates that the relationship between the horizontal factor \(H\) and the nozzle diameter \(d\) is linear.
02

Create Linear Function for Horizontal Factor

Let's set the diameter of the nozzle, \( d = 0.5 \) inches, with a corresponding horizontal factor \( H_0 = 56 \). For each \( \frac{1}{8} \) inch increase in \(d\), \(H\) increases by 6. We can express this as a linear function: \[ H(d) = 56 + 48 \times (d - 0.5) \] where \(48 = \frac{6}{\frac{1}{8}}\) is the number of \(1/8\)-inch increments in \(d\).
03

Calculate Horizontal Factor for Given Diameter

First, calculate the horizontal factor \(H\) when the nozzle diameter \( d = 1.75 \) inches: \[ H = 56 + 48 \times (1.75 - 0.5) \]\[ H = 56 + 48 \times 1.25 \]\[ H = 56 + 60 = 116\]
04

Calculate Distance with Pressure

With \(H = 116\) and pressure \(p = 50\), the distance \(S\) is: \[ S = \sqrt{116 \times 50} = \sqrt{5800} \approx 76.16 \text{ feet} \]
05

Calculate Horizontal Factor for Firefighter's Nozzle

Determine \(H\) for \(d = 1.25\) inches: \[ H = 56 + 48 \times (1.25 - 0.5) \]\[ H = 56 + 48 \times 0.75 \]\[ H = 56 + 36 = 92 \]
06

Calculate Distance for Firefighter's Scenario

Calculate \(S\) for \(H = 92\) and \(p = 70\): \[ S = \sqrt{92 \times 70} = \sqrt{6440} \approx 80.25 \text{ feet} \] Since \(S \approx 80.25\) feet is greater than 75 feet, the stream reaches the fire.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Water Pressure
Water pressure is an essential factor in determining how far a stream of water can travel from a hose. It is measured in pounds per square inch (psi) and provides the necessary force to propel water through a nozzle.
When considering the water pressure in a fire hose, it's crucial to understand that higher pressure results in a stronger and more forceful stream. Consequently, it allows the water to reach further distances.
With higher pressure, the particles of water move faster, which increases the stream's kinetic energy. Therefore, pressure directly influences the range a stream of water can achieve when the hose is held horizontally.
In the context of this exercise, the pressure of 50 psi is crucial to determine how far the water can travel. The formula used incorporates both the water pressure and the horizontal factor to calculate the stream's reach:
  • Higher pressure = more distance covered
  • Pressure works in tandem with horizontal factor
Understanding this concept helps firefighters ensure they have enough pressure to tackle fires from various distances effectively.
Horizontal Factor
The horizontal factor is a key component in understanding how far water can travel from a hose when held horizontally. It is essentially a measure that accounts for the effective reach of the nozzle based on its diameter.
The horizontal factor increases with the nozzle diameter, contributing to a longer stream. This factor is particularly important for situations like firefighting, where reaching great distances can be crucial.

How is the Horizontal Factor Linear?

A linear function describes the relationship between the horizontal factor and the nozzle diameter. This means that for every consistent increase in the nozzle diameter, the horizontal factor increases by a set amount.
For instance, in the given problem, the horizontal factor increases by 6 for every \( \frac{1}{8} \)-inch increase in diameter. This regularity in increase is what makes it linear. A linear function can be mathematically represented as:
  • Starting point: when diameter is 0.5 inches, the horizontal factor is 56
  • Increase: addition of 6 for every \( \frac{1}{8} \) inch
This straightforward relationship is captured in the linear equation provided: \[ H(d) = 56 + 48 \times (d - 0.5) \] Understanding this linearity allows for precise calculations in various practical scenarios.
Nozzle Diameter
The diameter of a nozzle plays a significant role in determining how far water can be projected from a hose. Working hand in hand with water pressure, the diameter affects the flow rate and the horizontal factor.
A larger nozzle diameter allows for a greater volume of water to pass through, thus enhancing the stream's reach. In contrast, a smaller diameter would restrict flow and decrease the distance.

Impact on Horizontal Factor

The nozzle diameter directly impacts the horizontal factor, which, as mentioned earlier, is linear. For each \( \frac{1}{8} \) inch increase in diameter, the horizontal factor rises by 6. This allows for more precise adjustments and calculations regarding how far the water will travel.

Real-World Application

For firefighters, choosing the right nozzle diameter based on the fire's location is crucial. By using the right combination of nozzle diameter and water pressure, they can ensure that the water will reach its target effectively.
In summary, understanding how the nozzle diameter affects the horizontal factor and water travel distance can lead to more efficient use of water resources in various practical applications.

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Most popular questions from this chapter

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