Answer :
The diameter of the pipe needed to carry a discharge of 500 gallons of water per minute at a velocity of 2 feet per second is approximately 35 inches.
To determine the diameter of a pipe needed to carry a discharge of 500 gallons of water per minute at a velocity of 2 feet per second, we can use the formula:
Q = (A * V)
Where:
Q is the flow rate (discharge) in gallons per minute
A is the cross-sectional area of the pipe in square inches
V is the velocity of the water in feet per second
First, let's convert the flow rate from gallons per minute to cubic inches per second:
Q = 500 gallons/minute * (1 minute/60 seconds) * (231 cubic inches/gallon)
Q = 1925 cubic inches/second
Next, let's rearrange the formula to solve for the cross-sectional area (A):
[tex]\begin{equation}A = \frac{Q}{V}[/tex]
Substituting the given values:
[tex]\begin{equation}A = \frac{1925\text{ in}^3/\text{s}}{2\text{ ft}/\text{s}}[/tex]
A = 962.5 square inches
Now, we can calculate the diameter (D) using the formula for the area of a circle:
[tex]\begin{equation}A = \pi \left(\frac{D}{2}\right)^2[/tex]
Rearranging the formula to solve for the diameter:
[tex]\begin{equation}D = \sqrt{\frac{4A}{\pi}}[/tex]
Substituting the value for A:
[tex]\[D = \sqrt{4 \times 962.5\text{ in}^2 / \pi} \\\\\approx \sqrt{3850 / 3.14159} \\\\\approx \sqrt{1225.015}\][/tex]
D ≈ 35 inches
Therefore, the diameter of the pipe needed to carry a discharge of 500 gallons of water per minute at a velocity of 2 feet per second is approximately 35 inches.
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