Explanation Given: A plane with equation x a + y b + z c = 1 ( a , b , c > 0 ) together with the positive co-ordinates planes forms a tetrahedron of volume V = 1 6 a b c . Calculation: Here. we need to find the minimum volume of V among all planes passing through the point P = ( α , β , γ ) . It is given that the equation of plane is x a + y b + z c = 1 and it passes through the point P = ( α , β , γ ) .So, the equation will be: α a + β b + γ c = 1 Let the function is V ( a , b , c ) = 1 6 a b c subjected to the plane α a + β b + γ c = 1 . The constraint curve is g ( a , b , c ) = α a + β b + γ c − 1 = 0 . We need to calculate ∇ V and ∇ g . Now, V ( a , b , c ) = 1 6 a b c ∇ V = ( V a ( a , b , c ) , V b ( a , b , c ) , V c ( a , b , c ) ) = ( ∂ ∂ a ( 1 6 a b c ) , ∂ ∂ b ( 1 6 a b c ) , ∂ ∂ c ( 1 6 a b c ) ) = ( 1 6 b c , 1 6 a c , 1 6 a b ) The constraint curve is g ( a , b , c ) = α a + β b + γ c − 1 . Then, g ( a , b , c ) = α a + β b + γ c − 1 ∇ g = ( g a ( a , b , c ) , g b ( a , b , c ) , g c ( a , b , c ) ) = ( ∂ ∂ a ( α a + β b + γ c − 1 ) , ∂ ∂ b ( α a + β b + γ c − 1 ) , ∂ ∂ c ( α a + β b + γ c − 1 ) ) = ( − α a 2 , − β b 2 , − γ c 2 ) Then the Lagrange equation is ∇ V = λ ∇ g ...

4th Edition

ISBN: 9781319379421

Author: Rogawski

Publisher: MAC HIGHER

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Question

Chapter 14.8, Problem 36E

To determine

The plane that minimizes V if the plane is constrained to pass through the point $P = (α, β, γ)$ with $α, β, γ > 0$ .

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Chapter 14.8, Problem 35E

Chapter 14.8, Problem 37E

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Chapter 14 Solutions

CALCULUS 4E (HC) W/ ACHIEVE ACCESS

Ch. 14.1 - Prob. 1PQ Ch. 14.1 - Prob. 2PQ Ch. 14.1 - Prob. 3PQ Ch. 14.1 - Prob. 4PQ Ch. 14.1 - Prob. 5PQ Ch. 14.1 - Prob. 1E Ch. 14.1 - Prob. 2E Ch. 14.1 - Prob. 3E Ch. 14.1 - Prob. 4E Ch. 14.1 - Prob. 5E

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The plane that minimizes V if the plane is constrained to pass through the point P = ( α , β , γ ) with α , β , γ &gt; 0 .

The plane that minimizes V if the plane is constrained to pass through the point P=(α,β,γ) with α,β,γ>0.

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Chapter 14 Solutions

The plane that minimizes V if the plane is constrained to pass through the point P = ( α , β , γ ) with α , β , γ > 0 .

The plane that minimizes V if the plane is constrained to pass through the point $P = (α, β, γ)$ with $α, β, γ > 0$ .