24. Oil Flow Oil flows through a cylindrical pipe of radius 3 inches, but friction from the pipe slows the flow toward the outer edge. The speed at which the oil flows at a distance r inches from the center is 8(10 - r²) inches per second. (a) In a plane cross section of the pipe, a thin ring with thick- ness Ar at a distance r inches from the center approximates a rectangular strip when you straighten it out. What is the area of the strip (and hence the approximate area of the ring)? (b) Explain why we know that oil passes through this ring at approximately 8(10-2) (2πr) Ar cubic inches per second. (c) Set up and evaluate a definite integral that will give the rate (in cubic inches per second) at which oil is flowing through the pipe.
24. Oil Flow Oil flows through a cylindrical pipe of radius 3 inches, but friction from the pipe slows the flow toward the outer edge. The speed at which the oil flows at a distance r inches from the center is 8(10 - r²) inches per second. (a) In a plane cross section of the pipe, a thin ring with thick- ness Ar at a distance r inches from the center approximates a rectangular strip when you straighten it out. What is the area of the strip (and hence the approximate area of the ring)? (b) Explain why we know that oil passes through this ring at approximately 8(10-2) (2πr) Ar cubic inches per second. (c) Set up and evaluate a definite integral that will give the rate (in cubic inches per second) at which oil is flowing through the pipe.
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Question
Transcribed Image Text:24. Oil Flow Oil flows through a cylindrical pipe of radius 3 inches,
but friction from the pipe slows the flow toward the outer edge.
The speed at which the oil flows at a distance r inches from the
center is 8(10 - r²) inches per second.
(a) In a plane cross section of the pipe, a thin ring with thick-
ness Ar at a distance r inches from the center approximates
a rectangular strip when you straighten it out. What is the
area of the strip (and hence the approximate area of the
ring)?
(b) Explain why we know that oil passes through this ring at
approximately 8(10-2) (2πr) Ar cubic inches per second.
(c) Set up and evaluate a definite integral that will give the rate
(in cubic inches per second) at which oil is flowing through
the pipe.
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