Asked by Elsi
Water is flowing in a cylindrical pipe of radius 1 inch. Because water is viscous and sticks to the pipe, the rate of flow varies with distance from the center. The speed of the water at distance r inches from the center is 10(1-(r^2)) per second. What is the rate (in cubic inches per second) at which water is flowing through the pipe?
Answers
Answered by
Steve
consider the amount of water that flows through a cross-section of the pipe in 1 second. It is a conical solid, high in the center and zero at the edges.
At radius r, the volume of the cylindrical shell of water is 2pi*r*h dr, where h = 10(1-r^2)
So, the amount across the whole cross-section is
integral[0,1] 2pi*r*10(1-r^2) dr
= 10pi r^2 - 5pi r^4 [0,1]
= 5pi in^3
since that's the volume in 1 second, the rate is 5pi in^3/s
At radius r, the volume of the cylindrical shell of water is 2pi*r*h dr, where h = 10(1-r^2)
So, the amount across the whole cross-section is
integral[0,1] 2pi*r*10(1-r^2) dr
= 10pi r^2 - 5pi r^4 [0,1]
= 5pi in^3
since that's the volume in 1 second, the rate is 5pi in^3/s
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