Problem 14 - Algebra 2

Question

The maximum flow of water in a pipe is modeled by the formula \(Q=A v,\) where \(A\) is the cross-sectional area of the pipe and \(v\) is the velocity of the water. Find the diameter of a pipe that allows a maximum flow of 50 \(\mathrm{ft}^{3} / \mathrm{min}\) of water flowing at a velocity of 600 \(\mathrm{ft} / \mathrm{min}\) . Round your answer to the nearest inch.

Step-by-Step Solution

Answer

The diameter of the pipe rounded to the nearest inch is 4 inches.

Step 1: Use given information to solve for the cross-sectional area A

The problem gives us \( Q = 50 \, \mathrm{ft}^3/\mathrm{min} \) and \( v = 600 \, \mathrm{ft}/\mathrm{min} \). We plug these values into the formula \( Q=A v \) to solve for \( A \). This gives us \( A = Q/v = 50/600 = 0.0833\, \mathrm{ft}^2 \).

Step 2: Solve for diameter d

We want to find the diameter \( d \) in terms of the area \( A \). The area of a circle is given by \( A= \pi (d/2)^2 \), so solving for \( d \) we get \( d= 2\sqrt{A/\pi}\). Substitute \( A = 0.0833 \, \mathrm{ft}^2 \) into the equation, resulting in \( d \approx 0.327 \, \mathrm{ft} \).

Step 3: Convert diameter to inches

As the problem requests the diameter to be in the nearest inch, we convert our result from feet to inches by using the conversion \( 1 \, \mathrm{ft} = 12\, \mathrm{in} \). This gives us \( d \approx 0.327 \times 12 \approx 3.92 \, \mathrm{in} \).

Key Concepts

Understanding the Cross-Sectional Area

When dealing with fluid dynamics, the cross-sectional area is an important parameter. It refers to the size of the surface through which fluid flows in a pipe. In most problems, like the one given, we deal with circular pipes. Here, the cross-sectional area can be calculated using the formula for the area of a circle: \(A = \pi (r^2)\), where \(r\) is the radius of the pipe.
This area helps determine the volume of fluid flowing through the pipe. If you know the velocity of the fluid, the cross-sectional area directly impacts the flow rate or maximum flow, calculated as \(Q = Av\). Understanding how to calculate and manipulate this measure is key for solving related problems in physics and engineering.

What is Velocity in This Context?

Velocity in the context of fluid dynamics is the speed at which fluid moves through a pipe. It is often measured in units such as feet per minute \(\mathrm{ft/min}\).
To calculate flow rate, the velocity is multiplied by the cross-sectional area, hence the formula \(Q = Av\). A higher velocity means a larger amount of fluid can flow in a set period of time, given a constant cross-sectional area.
In problems involving flow rate, knowing the velocity allows us to solve for the cross-sectional area or other unknowns like the diameter of the pipe.

How Pipe Diameter Affects Flow

The diameter of a pipe plays a crucial role in determining the cross-sectional area. Since the pipe is circular, the diameter directly affects both the flow rate and velocity of fluid in the pipe.
Given the formula \(A = \pi (d/2)^2\), you can see that even small changes in diameter can lead to significant changes in area. Moreover, larger diameters result in larger areas, which can allow more fluid to pass through; essential for designing pipes to accommodate desired flow rates.
For example, determining the correct diameter from a given flow rate and velocity involves rearranging the area formula to find diameter and then performing necessary calculations. This understanding is vital for various engineering applications.

The Role of Unit Conversion

Unit conversion is an essential skill when dealing with measurements in physics and engineering. Common conversions involve length measurements, like converting feet to inches or vice-versa.
In the given problem, after calculating the diameter in feet, it is necessary to convert the measurement to inches for a more understandable result. Using the conversion factor \(1 \, \mathrm{ft} = 12 \, \mathrm{in}\), you can easily transform feet into inches by multiplying. Hence, a diameter of \(0.327 \, \mathrm{ft}\) becomes \(3.92 \, \mathrm{in}\) when converted.
Always double-check conversions in your calculations to avoid errors and ensure that your results are in the desired unit of measurement.