Explanation Given: The cone height h and radius R is located at height h 4 on the central axis. Formula Used: ( x ¯ , y ¯ , z ¯ ) = ( 1 V ∫ ∫ W ∫ x d V , 1 V ∫ ∫ W ∫ y d V , 1 V ∫ ∫ W ∫ z d V ) Proof: The equation of the cone with height h and radius R is as: z = h r x 2 + y 2 0 ≤ r ≤ R , 0 ≤ θ ≤ 2 π , h r R ≤ z ≤ h The mass density is ρ ( x , y , z ) = 1 Then, the total mass of the cone is ∫ h R r h ∫ 0 2 π ∫ 0 R r d r d z d θ = ∫ 0 2 π ∫ 0 R [ z ] h R r h r d r d θ = ∫ 0 2 π ∫ 0 R ( h − h R r ) r d r d θ = ∫ 0 2 π ∫ 0 R ( h r − h r 2 R ) d r d θ = ∫ 0 2 π d θ [ h r 2 2 − h R r 3 3 ] 0 R = [ θ ] 0 2 π [ h R 2 2 − h R 2 3 ] = 1 3 π R 2 H Evaluating ∫ ∫ W ∫ x d V , ∫ ∫ W ∫ x d V = ∫ h R r h ∫ 0 2 π ∫ 0 R ( r cos θ ) r d z d r d θ = ∫ h R r h ∫ 0 2 π ∫ 0 R ( r 2 cos θ ) d z d r d θ = ∫ 0 2 π cos θ d θ ∫ 0 R ∫ h R r h r 2 d z d r = [ sin θ ] 0 2 π ∫ 0 R ∫ h R r h r 2 d z d r = 0 Evaluating ∫ ∫ W ∫ y d V , ∫ ∫ W ∫ y d V = ∫ h R r h ∫ 0 2 ...

4th Edition

ISBN: 9781319050733

Author: Rogawski

Publisher: MAC HIGHER

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Question

Chapter 16, Problem 46CRE

To determine

To prove:

The centroid of the right circular cone of height $h$ and radius $R$ is located at height $\frac{h}{4}$ on the central axis.

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Chapter 16, Problem 45CRE

Chapter 16, Problem 47CRE

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Chapter 16 Solutions

CALCULUS (CLOTH)

Ch. 16.1 - Prob. 1PQ Ch. 16.1 - Prob. 2PQ Ch. 16.1 - Prob. 3PQ Ch. 16.1 - Prob. 4PQ Ch. 16.1 - Prob. 5PQ Ch. 16.1 - Prob. 6PQ Ch. 16.1 - Prob. 1E Ch. 16.1 - Prob. 2E Ch. 16.1 - Prob. 3E Ch. 16.1 - Prob. 4E

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To prove: The centroid of the right circular cone of height h and radius R is located at height h 4 on the central axis.

Solution Summary: The author proves that the centroid of the right circular cone is located at height h 4 on the central axis.

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To prove:

The centroid of the right circular cone of height $h$ and radius $R$ is located at height $\frac{h}{4}$ on the central axis.

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Chapter 16 Solutions

To prove: The centroid of the right circular cone of height h and radius R is located at height h 4 on the central axis.

Solution Summary: The author proves that the centroid of the right circular cone is located at height h 4 on the central axis.

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To prove: The centroid of the right circular cone of height hand radius Ris located at height h4on the central axis.

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Chapter 16 Solutions

To prove:

The centroid of the right circular cone of height $h$ and radius $R$ is located at height $\frac{h}{4}$ on the central axis.