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Chapter 2: Problem 9

For firefighting an important concept is that of engine pressure, which is the pressure that the pumper requires to overcome friction loss from the hose and to maintain the proper pressure at the nozzle of the hose. One formula \({ }^{14}\) for engine pressure is $$ E P=N P(1.1+K \times L), $$ where \(E P\) is the engine pressure in pounds per square inch (psi), \(N P\) is the nozzle pressure in psi, \(K\) is the " \(K\) " factor (which depends on which tip is used on the nozzle), and \(L\) is the length of the hose in feet divided by 50 . This formula expresses \(E P\) as a function of \(N P, K\), and \(L\). a. Find the engine pressure if the nozzle pressure is 80 psi, the " \(\mathrm{K}\) " factor is \(0.51\), and the length of the hose is 80 feet. b. Find the nozzle pressure if the " \(\mathrm{K}\) " factor is \(0.73\), the length of the hose is 45 feet, and the engine pressure is \(150 \mathrm{psi}\). c. Solve the equation for \(N P\)-that is, write a formula expressing \(N P\) as a function of \(E P, K\), and \(L\). d. Find the " \(K\) " factor for a nozzle tip if the length of the hose is 190 feet, the engine pressure is \(160 \mathrm{psi}\), and the nozzle pressure is \(90 \mathrm{psi}\). e. Solve the equation for \(K\)-that is, write a formula expressing \(K\) as a function of \(N P, E P\), and \(L\).

Short Answer

Expert verified
a) EP = 153.28 psi, b) NP = 85.36 psi, c) \( NP = \frac{EP}{1.1 + K \times L} \), d) K 鈮� 0.17863, e) \( K = \frac{\frac{EP}{NP} - 1.1}{L} \).

Step by step solution

01

Calculate L for Part a

Given the hose length is 80 feet, calculate \( L \) as \( \frac{80}{50} = 1.6 \).
02

Calculate EP for Part a

Substitute \( NP = 80 \), \( K = 0.51 \), and \( L = 1.6 \) into the formula: \[ EP = 80(1.1 + 0.51 \times 1.6) \] Calculate \( 0.51 \times 1.6 = 0.816 \); hence, \[ EP = 80(1.916) = 153.28 \].
03

Solve for L in Part b

Given the hose length of 45 feet, calculate \( L \) as \( \frac{45}{50} = 0.9 \).
04

Rearrange Formula for NP in Part b

Substitute \( K = 0.73 \) and \( L = 0.9 \), and set \( EP = 150 \) to solve for \( NP \): \( 150 = NP(1.1 + 0.73 \times 0.9) \).
05

Calculate NP for Part b

Calculate \( 0.73 \times 0.9 = 0.657 \); hence, \( 150 = NP(1.757) \). Divide both sides by \( 1.757 \): \( NP = \frac{150}{1.757} = 85.36 \).
06

Express NP as a Function in Part c

Rearrange the original formula to solve for \( NP \): \[ NP = \frac{EP}{1.1 + K \times L} \].
07

Calculate L for Part d

Given the hose length is 190 feet, calculate \( L \) as \( \frac{190}{50} = 3.8 \).
08

Rearrange Formula for K in Part d

Set \( EP = 160 \), \( NP = 90 \), and \( L = 3.8 \) in the formula, \( 160 = 90(1.1 + K \times 3.8) \).
09

Solve for K in Part d

Divide both sides by 90: \( 1.7778 \approx 1.1 + K \times 3.8 \) Subtract 1.1: \( 0.6778 \approx K \times 3.8 \) Divide by 3.8: \( K \approx 0.17863 \).
10

Express K as a Function in Part e

Rearrange the original formula to solve for \( K \): \[ K = \frac{\frac{EP}{NP} - 1.1}{L} \].

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Key Concepts

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

Functions
In mathematics, functions are crucial concepts that describe relationships between variables. In this exercise, the function represents engine pressure (56) as a relationship involving nozzle pressure (56), the K factor (b), and the length of the hose (c). The equation is: \[ EP = NP (1.1 + K \times L) \]. The nozzle pressure is the pressure exerted by the water as it exits the nozzle. The K factor depends on the nozzle tip design and affects how friction impacts pressure. The length of the hose, divided by 50, scales based on hose length's impact on pressure loss. Understanding functions is key to solving any problems arising from changing variables and determining how one variable influences another. In this exercise, knowing how to rearrange the formula to solve for different variables, such as \( NP \) or \( K \), reveals the flexible nature of functions.
Modeling
Modeling in mathematics involves creating equations or formulas that represent real-world situations. Here, the engine pressure formula models how various firefighting conditions affect the pressure required to operate a hose effectively. This formula encapsulates the interaction between friction loss, nozzle pressure, and hose length, integrated by the K factor, which indicates how different nozzle tips influence the pressure dynamics. Mathematical models like this are powerful in predictive scenarios, allowing for calculations of unknown parameters when certain variables are known. They serve as tools to prepare and ensure safety in practical fields like firefighting, where pressure management is crucial. In this exercise, modeling aids in determining the necessary conditions to achieve desired water pressure at the nozzle, taking into account differing hose lengths and apparatus settings.
Problem Solving
Problem solving in algebra is the process of using mathematical principles to find solutions. This exercise tasks you with manipulating the engine pressure equation to find unknown variables under different scenarios. Consider these steps:
  • Identify known and unknown factors in the equation.
  • Rearrange the formula to solve for the desired variable.
  • Substitute known values into the equation.
  • Perform arithmetic operations to find the solution.
Through methodical problem solving, we derive how different elements like nozzle pressure, engine pressure, and the K factor interact, showcasing the dynamic and interdependent nature of these systems. Problem solving empowers you to predict outcomes and understand system behaviors in real-world applications.
Pressure Calculations
Calculating pressure in this context involves understanding the variables affecting it within firefighting systems. Pressure is fundamentally about force over an area. In the formula \( EP = NP (1.1 + K \times L) \), the pressure exerted by the engine must counteract friction losses and ensure adequate pressure at the nozzle. This exercise involves:
  • Calculating equivalent hose length \( L \) by dividing actual length by 50.
  • Incorporating \( K \) factors, which adjust calculations based on nozzle design.
  • Using the formula to calculate \( EP \) for different scenarios, thus solving for different unknowns.
Engaging with pressure calculations enhances comprehension of hydraulic systems in firefighting, vital for ensuring effective operation and safety under various conditions.

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