Problem 57 - Algebra 1

Question

The formula \(d=\frac{n}{2}+26\) relates nozzle pressure \(n\) (in pounds per square inch) and the maximum distance the water reaches \(d\) (in feet) for a fire hose with a certain size nozzle. How much pressure is needed if such a hose is held 50 feet from a fire?

Step-by-Step Solution

Answer

The nozzle pressure needed if the fire hose is 50 feet from the fire is 48 pounds per square inch

Step 1: Understand the problem

Identify that we are given the desired maximum distance (\(d = 50\) feet), and we need to find the corresponding pressure \(n\). Recall the formula: \(d=\frac{n}{2}+26\).

Step 2: Rearrange the equation

First, isolate \(n\) in the equation. This can be done by subtracting 26 from both sides and then multiplying both sides by 2. This gives: \(n = 2(d - 26)\).

Step 3: Substitute and solve

Substitute \(d = 50\) into the rearranged equation to find \(n\):\n\(n = 2(50 - 26) = 2(24) = 48\).

Key Concepts

Nozzle Pressure

When tackling problems involving nozzle pressure, it's important to understand the role of pressure in water distribution systems like hoses and nozzles. Nozzle pressure, measured in pounds per square inch (psi), is how hard the water is being pushed through the nozzle.
In this context, higher nozzle pressure will result in the water reaching a greater distance, making it crucial during firefighting to control how far water can be sprayed. This calculation helps ensure that water can reach the fire efficiently without too much loss of flow.

Distance Calculation

Calculating the distance water reaches when ejected from a nozzle involves understanding the relationship between pressure and distance. In the given formula\(d = \frac{n}{2} + 26\), we see that the distance \(d\) is directly affected by the nozzle pressure \(n\).
For every 2 psi increase in pressure, the distance increases by one foot, which means nozzle pressure has a significant impact on how far water can be projected. By substituting different values into the formula, you can calculate the maximum reach distance for a given pressure, which helps in planning effective firefighting strategies.

Rearranging Formulas

Rearranging formulas is a key algebraic skill that involves manipulating equations to express one variable in terms of others. In our example, we need to isolate the nozzle pressure \(n\) in the formula to find its value for a specific distance.
Here's how you do it:

• Given the formula: \(d = \frac{n}{2} + 26\)
• Subtract 26 from both sides to isolate the fraction: \(d - 26 = \frac{n}{2}\)
• Multiply both sides by 2 to solve for \(n\): \(n = 2(d - 26)\)

By following these steps, you can easily rearrange the original formula to solve for nozzle pressure when a specific distance is required. This algebraic manipulation is crucial for using mathematical formulas effectively in practical situations.