To determine Determine the point where radio telescope is located. Explanation Given information: The magnetic field is expressed as M ( x , y , z ) = 6 x − y 2 + x z + 60 . The origin is at the center of the planet with the radius of 6 units. Calculation: Show the magnetic field at ( x , y , z ) as follows: M ( x , y , z ) = 6 x − y 2 + x z + 60 (1) Find the gradient ( ∇ M ) using Equation (1). ∇ M = 6 i − 2 y j + z i + x k = ( 6 + z ) i − 2 y j + x k Show the constraint as origin is at the center with radius of 4 units as follows: g ( x , y , z ) = x 2 + y 2 + z 2 − 36 = 0 (2) Find the gradient ( ∇ g ) using Equation (2). ∇ g = 2 x i + 2 y j + 2 z k Find the values of x , y , z , and λ that satisfies the following equation: ∇ M = λ ∇ g Substitute ( 6 + z ) i − 2 y j + x k for ∇ M and 2 x i + 2 y j + 2 z k for ∇ g . ( 6 + z ) i − 2 y j + x k = λ ( 2 x i + 2 y j + 2 z k ) ( 6 + z ) = 2 x λ , − 2 y = 2 y λ , and x = 2 z λ Simplify − 2 y = 2 y λ as follows: − 2 y = 2 y λ λ = − 1 or y = 0 Case 1: If the value of λ = − 1 , then ( 6 + z ) = 2 x λ becomes ( 6 + z ) = 2 x ( − 1 ) 6 + z = − 2 x And x = 2 z λ becomes, x = 2 z ( − 1 ) x = − 2 z Substitute − 2 z for x in 6 + z = − 2 x . 6 + z = − 2 ( − 2 z ) z = 2 Substitute 2 for z in x = − 2 z . x = − 2 ( 2 ) = − 4 Substitute − 4 for x and 2 for z in Equation (1). ( − 4 ) 2 + y 2 + 2 2 = 36 y 2 = 16 y = ± 4 Case 2: If the value of y = 0 , then x = 2 z λ becomes λ = x 2 z Substitute x 2 z for λ in ( 6 + z ) = 2 x λ

Thomas' Calculus: Early Transcendentals, Single Variable (14th Edition)

14th Edition

ISBN: 9780134439419

Author: Joel R. Hass, Christopher E. Heil, Maurice D. Weir

Publisher: PEARSON

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Chapter 14.8, Problem 36E

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Determine the point where radio telescope is located.

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Chapter 14.8, Problem 35E

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