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The discharge of a fire hose depends on the diameter of the nozzle. Nozzle diameters are normally in multiples of \(\frac{1}{8}\) inch. Sometimes it is important to replace several hoses with a single hose of equivalent discharge capacity. Hoses with nozzle diameters \(d_{1}, d_{2}, \ldots, d_{n}\) have the same discharge capacity as a single hose with nozzle diameter \(D\), where $$ D=\sqrt{d_{1}^{2}+d_{2}^{2}+\cdots+d_{n}^{2}} . $$ a. A nozzle of what diameter has the same discharge capacity as three combined nozzles of diameters \(1 \frac{1}{8}\) inches, \(1 \frac{5}{8}\) inches, and \(1 \frac{7}{8}\) inches? You should report your answer as an available nozzle size, that is, in multiples of \(\frac{1}{8}\). b. We have two 1 -inch nozzles and wish to use a third so that the combined discharge capacity of the three nozzles is the same as the discharge capacity of a \(2 \frac{1}{4}\)-inch nozzle. What should be the diameter of the third nozzle? c. If we wish to use \(n\) hoses each with nozzle size \(d\) in order to have the combined discharge capacity of a single hose with nozzle size \(D\), then we must use $$ n=\left(\frac{D}{d}\right)^{2} \text { nozzles. } $$ How many half-inch nozzles are needed to attain the discharge capacity of a 2 -inch nozzle? d. We want to replace a nozzle of diameter \(2 \frac{1}{4}\) inches with 4 hoses each of the same nozzle diameter. What nozzle diameter for the 4 hoses will produce the same discharge capacity as the single hose?

Step by step solution

01

Convert Mixed Numbers to Improper Fractions

First, convert the given mixed numbers into improper fractions. For the nozzle diameters of \(1 \frac{1}{8}\), \(1 \frac{5}{8}\), and \(1 \frac{7}{8}\) inches, the improper fractions are \(\frac{9}{8}\), \(\frac{13}{8}\), and \(\frac{15}{8}\) respectively.

02

Apply the Discharge Formula

Substitute the improper fractions into the discharge capacity formula:\[D = \sqrt{\left(\frac{9}{8}\right)^2 + \left(\frac{13}{8}\right)^2 + \left(\frac{15}{8}\right)^2}.\]

03

Calculate Each Term's Square

Calculate each term's square:- \(\left(\frac{9}{8}\right)^2 = \frac{81}{64}\)- \(\left(\frac{13}{8}\right)^2 = \frac{169}{64}\)- \(\left(\frac{15}{8}\right)^2 = \frac{225}{64}\).

04

Sum the Squared Terms

Add the squared terms:\[\frac{81}{64} + \frac{169}{64} + \frac{225}{64} = \frac{475}{64}.\]

05

Calculate Square Root for Total Diameter

Find the square root to get the total diameter:\[D = \sqrt{\frac{475}{64}} = \frac{\sqrt{475}}{8}.\]

06

Convert Diameter to Nearest Multiple of 1/8

Estimate \(\sqrt{475} \approx 21.8\), hence \(D = \frac{21.8}{8} \approx 2.725\), closest to \(2 \frac{5}{8} = \frac{21}{8}\) inches.

07

Solve for Diameter of Third Nozzle

Use the discharge formula to solve for the third nozzle:\[2.25 = \sqrt{1^2 + 1^2 + d^2}\]Simplify to solve:\[2.25^2 = 1^2 + 1^2 + d^2 \5.0625 = 2 + d^2 \d^2 = 3.0625 \d = \sqrt{3.0625} \approx 1.75 = \frac{14}{8} = 1 \frac{6}{8}\]

08

Calculate Number of Nozzles for Equivalent Discharge

Substitute into the formula for number of nozzles:\[n = \left(\frac{2}{\frac{1}{2}}\right)^2 = (4)^2 = 16\] half-inch nozzles are needed.

09

Diameter for Four Equivalent Hoses

Apply the formula to determine the equivalent nozzle diameter for four hoses:\[2.25 = \sqrt{4 \cdot d^2} \2.25 = 2d \d = 1.125 = \frac{9}{8} \text{ inches}\]

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Key Concepts

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

Discharge Capacity

Discharge capacity refers to the volume of a fluid that a nozzle can expel over a given period of time. In simpler terms, it's how much water, for example, can flow out of a nozzle. When dealing with firefighting equipment like hoses, understanding discharge capacity is crucial because it influences the effectiveness of a hose in delivering water to extinguish fires.
To calculate the discharge capacity when multiple nozzles are involved, we use a formula that gives us the "equivalent" diameter. This tells us what single nozzle diameter would match the combined discharge capacity of several smaller nozzles.

• This formula is: \[ D = \sqrt{d_1^2 + d_2^2 + \cdots + d_n^2} \]
• "D" represents the diameter of a single equivalent nozzle.
• Each "d" stands for the diameters of each individual nozzle.

Understanding and applying this formula allows for more efficient planning and resource management, especially in situations where performance and speed are essential.

Nozzle Diameter

Nozzle diameter is a critical measurement in determining how much fluid a nozzle can discharge. The diameter directly impacts the rate and pressure at which fluid is expelled. Nozzles in firefighting, for example, typically have diameters measured in fractions, such as \(\frac{1}{8}\) inch or \(1 \frac{1}{8}\) inches.
When you need to replace multiple nozzles with a single nozzle of equivalent capacity, it's crucial to understand the precise measurements. To convert mixed numbers into improper fractions:

• \(1 \frac{1}{8} = \frac{9}{8}\)
• \(1 \frac{5}{8} = \frac{13}{8}\)
• \(1 \frac{7}{8} = \frac{15}{8}\)

Using these fractions in the discharge capacity formula helps determine the required equivalent nozzle diameter. Precise calculations ensure safety and efficiency in demanding situations.

Equivalent Nozzles

Equivalent nozzles are used to replace several nozzles with one that has the same discharge capacity. Finding equivalent nozzles is often needed to simplify setups, manage resources, or meet specific requirements in equipment functionality.
To find an equivalent nozzle diameter when replacing several nozzles:

• Convert each diameter to a suitable fraction or decimal.
• Use the formula: \[ D = \sqrt{d_1^2 + d_2^2 + \cdots + d_n^2} \]

This formula helps in determining the single nozzle diameter (D) capable of the same fluid disbursement rate as multiple smaller nozzles. Practical applications include firefighting and irrigation, where nozzle interchangeability can aid in achieving the desired water flow with fewer setup changes.

Mathematical Problem-Solving

Mathematical problem-solving involves various techniques to find solutions to equations and real-world scenarios, accurately and efficiently. In our nozzle exercise, problem-solving starts with understanding and converting units or dimensions into suitable forms for calculations, like improper fractions or decimals.
For problems involving nozzle diameters:

• Begin by converting mixed numbers into improper fractions for accuracy.
• Apply the correct formula, substituting known values for simplification.
• Carry out step-by-step calculations — compute powers, add terms, and find roots.

For instance, when determining how many smaller nozzles are needed to replace a larger one, you'd apply the formula:\[ n = \left(\frac{D}{d}\right)^2 \] This involves comparing the diameters, finding how many smaller nozzles (\"d\") are equivalent to a larger single nozzle (\(D\)). By carefully following and applying these problem-solving steps, solutions become manageable and understandable.