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You may have seen those "dumping buckets" at a pool or water park. There is a water source filling a bucket. At some point when the bucket becomes full it will suddenly tilt, splashing peo- ple beneath. The idea of this worksheet is to analyze the physics of the dumping bucket us- ing calculus. For purposes of this problem the bucket will be assumed to have a height of 0.5 m. The vertical coordinate will be denoted by æ, and the pivot axis for the bucket is halfway up, at height a = 0.25 m. The cross-sectional ra- dius at bottom, x = 0, is 0.125m and the cross- sectional radius at top, r = 0.5m, is 0.5m. Our idealized bucket. Problem 1 (The Bucket). a) If the bucket is a section of a circular cone then the radius of the cross-section grows linearly with height. Find an expression for the radius as a function of height. b) Using the formula V = S A(y)dy, where A(y) is the cross-sectional area at height y, find the total volume of the bucket in liters.

You may have seen those "dumping buckets" at a pool or water park. There is a water source filling a bucket. At some point when the bucket becomes full it will suddenly tilt, splashing peo- ple beneath. The idea of this worksheet is to analyze the physics of the dumping bucket us- ing calculus. For purposes of this problem the bucket will be assumed to have a height of 0.5 m. The vertical coordinate will be denoted by æ, and the pivot axis for the bucket is halfway up, at height a = 0.25 m. The cross-sectional ra- dius at bottom, x = 0, is 0.125m and the cross- sectional radius at top, r = 0.5m, is 0.5m. Our idealized bucket. Problem 1 (The Bucket). a) If the bucket is a section of a circular cone then the radius of the cross-section grows linearly with height. Find an expression for the radius as a function of height. b) Using the formula V = S A(y)dy, where A(y) is the cross-sectional area at height y, find the total volume of the bucket in liters.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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You may have seen those "dumping buckets" at a pool or water park. There is a water source filling a bucket. At some point when the bucket becomes full it will suddenly tilt, splashing peo- ple beneath. The idea of this worksheet is to analyze the physics of the dumping bucket us- ing calculus. For purposes of this problem the bucket will be assumed to have a height of 0.5 m. The vertical coordinate will be denoted by x, and the pivot axis for the bucket is halfway up, at height x = 0.25 m. The cross-sectional ra- dius at bottom, x = 0, is 0.125m and the cross- sectional radius at top, x = 0.5m, is 0.5m. Our idealized bucket. Problem 1 (The Bucket). a) If the bucket is a section of a circular cone then the radius of the cross-section grows linearly with height. Find an expression for the radius as a function of height. b) Using the formula V = [ A(y)dy, where A(y) is the cross-sectional area at height y, find the total volume of the bucket in liters.
Transcribed Image Text:You may have seen those "dumping buckets" at a pool or water park. There is a water source filling a bucket. At some point when the bucket becomes full it will suddenly tilt, splashing peo- ple beneath. The idea of this worksheet is to analyze the physics of the dumping bucket us- ing calculus. For purposes of this problem the bucket will be assumed to have a height of 0.5 m. The vertical coordinate will be denoted by x, and the pivot axis for the bucket is halfway up, at height x = 0.25 m. The cross-sectional ra- dius at bottom, x = 0, is 0.125m and the cross- sectional radius at top, x = 0.5m, is 0.5m. Our idealized bucket. Problem 1 (The Bucket). a) If the bucket is a section of a circular cone then the radius of the cross-section grows linearly with height. Find an expression for the radius as a function of height. b) Using the formula V = [ A(y)dy, where A(y) is the cross-sectional area at height y, find the total volume of the bucket in liters.
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