Explanation Given: z-coordinate z ¯ = c 4 . The tetrahedron bounded by the coordinate planes and the plane x a + y b + z c = 1 . Formula Used: Centroid is given by: ( x ¯ , y ¯ , z ¯ ) = ( 1 V ∭ W x d V , 1 V ∭ W y d V , 1 V ∭ W z d V ) ... ( 1 ) Proof: Assume, density δ ( x , y ) = 1 . Here, the integration limits will be: 0 ≤ x ≤ a , 0 ≤ y ≤ b ( 1 − x a ) and 0 ≤ z ≤ c ( 1 − x a − y b ) . First evaluate V = ∭ W δ ( x , y , z ) d V and we get: V = ∭ W δ ( x , y , z ) d V = ∫ x = 0 a ∫ y = 0 b ( 1 − x a ) ∫ z = 0 c ( 1 − x a − y b ) d z d y d x = ∫ x = 0 a ∫ y = 0 b ( 1 − x a ) [ z ] 0 c ( 1 − x a − y b ) d y d x = ∫ x = 0 a ∫ y = 0 b ( 1 − x a ) c ( 1 − x a − y b ) d y d x = c ∫ 0 a [ ( 1 − x a ) y − 1 2 b y 2 ] 0 b ( 1 − x a ) d x = c ∫ 0 a [ ( 1 − x a ) b ( 1 − x a ) − 1 2 b ( b ( 1 − x a ) ) 2 ] d x = c b 2 ∫ 0 a ( 1 − x a ) 2 d x = c b 2 [ − ( a − x ) 3 3 a 2 ] 0 a = c b 2 ( a 3 ) = a b c 6 ...

3rd Edition

ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

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Chapter 16.5, Problem 20E

To determine

To Prove:

Thez-coordinate of the centroid of the tetrahedron bounded by the coordinate planes and the plane $\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1$ in figure 14 is $\bar{z} = \frac{c}{4}$ . Conclude by symmetry that the centroid is $(\frac{a}{4}, \frac{b}{4}, \frac{c}{4})$ .

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Chapter 16.5, Problem 19E

Chapter 16.5, Problem 21E

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Chapter 16 Solutions

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To Prove: Thez-coordinate of the centroid of the tetrahedron bounded by the coordinate planes and the plane x a + y b + z c = 1 in figure 14 is z ¯ = c 4 . Conclude by symmetry that the centroid is ( a 4 , b 4 , c 4 ) .

Solution Summary: The author concludes by symmetry that the centroid of the tetrahedron bounded by the coordinate planes is z = c 4.

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To Prove:

Thez-coordinate of the centroid of the tetrahedron bounded by the coordinate planes and the plane $\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1$ in figure 14 is $\bar{z} = \frac{c}{4}$ . Conclude by symmetry that the centroid is $(\frac{a}{4}, \frac{b}{4}, \frac{c}{4})$ .

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Chapter 16 Solutions

To Prove: Thez-coordinate of the centroid of the tetrahedron bounded by the coordinate planes and the plane x a + y b + z c = 1 in figure 14 is z ¯ = c 4 . Conclude by symmetry that the centroid is ( a 4 , b 4 , c 4 ) .

Solution Summary: The author concludes by symmetry that the centroid of the tetrahedron bounded by the coordinate planes is z = c 4.

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To Prove: Thez-coordinate of the centroid of the tetrahedron bounded by the coordinate planes and the plane xa+yb+zc=1in figure 14 is z¯=c4. Conclude by symmetry that the centroid is (a4,b4,c4).

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Chapter 16 Solutions

To Prove:

Thez-coordinate of the centroid of the tetrahedron bounded by the coordinate planes and the plane $\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1$ in figure 14 is $\bar{z} = \frac{c}{4}$ . Conclude by symmetry that the centroid is $(\frac{a}{4}, \frac{b}{4}, \frac{c}{4})$ .