Question

1. a pipe has a diameter of 48 inches and is full of water. if the velocity through the pipe is 20 feet per minute, what is the flow rate through the pipe in gallons per minute?

Explanation:

Step1: Convert diameter to radius in feet

The diameter of the pipe is 48 inches. Since 1 foot = 12 inches, the radius \( r=\frac{48\div12}{2}=\frac{4}{2} = 2\) feet.

Step2: Calculate the cross - sectional area of the pipe

The cross - sectional area \( A\) of a circle is given by the formula \( A=\pi r^{2}\). Substituting \( r = 2\) feet, we get \( A=\pi\times(2)^{2}=4\pi\) square feet.

Step3: Calculate the volume flow rate in cubic feet per minute

The volume flow rate \( Q\) (in cubic feet per minute) is given by the product of the cross - sectional area \( A\) and the velocity \( v\). The velocity \( v = 20\) feet per minute. So \( Q=A\times v=4\pi\times20 = 80\pi\) cubic feet per minute.

Step4: Convert cubic feet per minute to gallons per minute

We know that 1 cubic foot is approximately equal to 7.48052 gallons. So the flow rate in gallons per minute \( Q_{gallons}=80\pi\times7.48052\).
\( Q_{gallons}\approx80\times3.1416\times7.48052\)
First, \( 80\times3.1416 = 251.328\)
Then, \( 251.328\times7.48052\approx1879.23\)

Answer:

The flow rate through the pipe is approximately \(\boldsymbol{1879.23}\) gallons per minute.