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Vertical reach of fire hoses: If a fire hose is held vertically, then the height the stream will travel depends on water pressure and on the vertical factor for the nozzle. The vertical factor \(V\) depends on the diameter of the nozzle. For a \(0.5\)-inch nozzle, the vertical factor is 85 . For each \(\frac{1}{8}\)-inch increase in nozzle diameter, the vertical factor increases by 5 . a. Explain why the function giving the vertical factor \(V\) in terms of the nozzle diameter \(d\) is linear. b. Use a formula to express \(V\) as a linear function of \(d\) (measured in inches). c. Once the vertical factor is known, we can calculate the height \(S\) in feet that a vertical stream of water can travel by using $$ S=\sqrt{V p} \text {. } $$ Here \(p\) is pressure in pounds per square inch. How high will a vertical stream travel if the pressure is 50 pounds per square inch and the nozzle diameter is \(1.75\) inches? d. Firemen have a nozzle with a diameter of \(1.25\) inches. The pumper generates a pressure of 70 pounds per square inch. From street level, they need to get water on a fire 60 feet overhead. Can they reach the fire with a vertical stream of water?

Step by step solution

01

Define the linear function

The vertical factor for a 0.5-inch nozzle is given as 85. With each 1/8-inch increase in diameter, the vertical factor increases by 5 units. This means that the relationship between the diameter and the vertical factor is linear, as it affects by a constant amount (5) for each increment of 1/8-inch.

02

Determine the linear equation

To express the vertical factor \( V \) as a linear function of the diameter \( d \), let's consider:1. For \( d = 0.5 \), \( V = 85 \).2. Each 1/8 inch increment in \( d \) leads to a 5 unit increment in \( V\).Since a 1/8 inch increment changes \( V \) by 5, the slope is \( \frac{5}{1/8} = 40 \). Therefore, the linear function is:\[ V = 40d + b \]Plug in \( d = 0.5 \) and \( V = 85 \) to find \( b \):\[ 85 = 40(0.5) + b \]\[ 85 = 20 + b \]\[ b = 65 \]Thus, \( V = 40d + 65 \).

03

Calculate the stream height for given conditions

For part c, with a pressure \( p = 50 \) psi and a nozzle diameter \( d = 1.75 \) inches:Find \( V \) using \( V = 40d + 65 \): \[ V = 40(1.75) + 65 = 70 + 65 = 135 \]Now calculate \( S \) using \( S = \sqrt{Vp} \):\[ S = \sqrt{135 \times 50} = \sqrt{6750} \approx 82.15 \] feet.

04

Assess the capability with current nozzle and pressure

For part d, with a pressure \( p = 70 \) psi and a nozzle diameter \( d = 1.25 \) inches:Find \( V \) using \( V = 40d + 65 \):\[ V = 40(1.25) + 65 = 50 + 65 = 115 \]Calculate \( S \) using \( S = \sqrt{Vp} \):\[ S = \sqrt{115 \times 70} = \sqrt{8050} \approx 89.73 \] feet.Since \( S = 89.73 \) feet is greater than 60 feet, the firemen can reach the fire with the vertical stream of water.

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Key Concepts

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

Understanding Nozzle Diameter

When we talk about nozzle diameter, we're referring to the width of the opening through which water exits the nozzle. This diameter is a crucial factor in determining how water is dispensed from the hose. A larger diameter means more water can flow out, but this affects the water pressure and its reach.

In our exercise, the nozzle diameter directly affects the vertical factor, which describes how high water from the hose can travel. For instance, with a standard nozzle diameter of 0.5 inches, we have a baseline vertical factor of 85. As the nozzle diameter increases by increments of 1/8 inches, the vertical factor increases by 5. This clearly shows a linear relationship between the nozzle diameter and the vertical factor. Understanding this concept is essential, especially in firefighting, where efficient water delivery can be the difference in controlling a blaze.

Exploring Vertical Factor

The vertical factor is an interesting concept that relates to how high a stream of water can be projected vertically. In our scenario, the vertical factor varies as the nozzle diameter changes, displaying a linear relationship.

The vertical factor is crucial for determining how high a stream can reach, particularly in firefighting, where reaching higher elevations with water is often necessary. For a nozzle of 0.5-inch diameter, the vertical factor is 85. With each 1/8-inch diameter increase in the nozzle, the vertical factor increases by 5. This consistent change allows us to describe the relationship with a linear function:

• Start point: 0.5 inches, factor 85
• Increment: 1/8 inch leads to a 5 unit increase in factor

This predictability is invaluable for firefighters planning their equipment use.

Fire Hose Dynamics and Their Implications

Fire hose dynamics cover a range of factors affecting how water is delivered from the hose. It involves understanding the interaction between nozzle diameter, water pressure, and the vertical factor. The linear relationship between nozzle diameter and vertical factor is just one aspect of these dynamics.

• A larger nozzle diameter can increase the amount of water delivered but may lower the height due to decreased pressure per unit area.
• The optimal balance between diameter and water pressure ensures efficient delivery.
• Knowing these dynamics helps firefighters adjust strategies, ensuring that they can reach targets effectively and conserve water resources during a fire.

By understanding fire hose dynamics, teams can better plan their attack strategies against fires and optimize resource usage.

All About Water Pressure

Water pressure is a force that pushes water through the hose, and it's measured in pounds per square inch (psi). It's a vital component in calculating how far the water can travel. When working with fire hoses, this pressure directly influences the height that the water stream can reach.

The formula to calculate height based on the vertical factor and water pressure is:

• \[ S = \sqrt{V \cdot p} \]

In this formula, \( S \) is the stream height, \( V \) is the vertical factor, and \( p \) is the pressure. If you increase the pressure, the potential stream height increases too, which is crucial in reaching fires at elevated positions. By balancing the pressure correctly with the vertical factor, firefighters can maximize efficiency and ensure water reaches the necessary height to combat fires successfully.