HW_12_Key

# HW_12_Key - Version 084 – Homework 12 – Gilbert...

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Unformatted text preview: Version 084 – Homework 12 – Gilbert – (59825) 1 This print-out should have 11 questions. Multiple-choice questions may continue on the next column or page – find all choices before answering. The due time is Central time. I’ve put a few numerical free response questions on this assignment to test this feature. Return your answer as a deci- mal to, say, 3 places. For the answer to be graded as correct is has to be within 1% of the correct answer. 001 10.0 points Determine the minimum value of f ( x, y, z ) = 2 x 2 + y 2 + 2 z 2 + 2 subject to the constraint 3 x + 2 y + 2 z = 5 . 1. min value = 30 7 2. min value = 32 7 3. min value = 89 21 4. min value = 92 21 correct 5. min value = 94 21 Explanation: By the method of Lagrange Multipliers the minimum value of f ( x, y, z ) = 2 x 2 + y 2 + 2 z 2 + 2 subject to the constraint ( † ) 3 x + 2 y + 2 z = 5 will occur at a critical point of the Lagrange function F ( x, y, z, λ ) = 2 x 2 + y 2 + 2 z 2 + 2 + λ (3 x + 2 y + 2 z- 5) . This critical point will be the common solu- tion ( x , y , z , λ ) of the equations ∂F ∂x = 4 x + 3 λ = 0 , ∂F ∂y = 2 y + 2 λ = 0 , ∂F ∂z = 4 z + 2 λ = 0 , and ∂F ∂λ = 3 x + 2 y + 2 z- 5 = 0 . Solving, we see that x =- 3 4 λ , y =- λ , z =- 1 2 λ , so 21 4 λ + 5 = 0 , i.e. , λ =- 20 21 . Thus the minimum value of f subject to the constraint ( † ) occurs at parenleftBig 5 7 , 20 21 , 10 21 parenrightBig . Consequently, the minimum value of f subject to the constraint ( † ) is given by min value = 92 21 . 002 10.0 points The temperature T at a point ( x, y, z ) on the surface x 2 + y 2 + z 2 = 48 is given by T ( x, y, z ) = x + y + z in degrees centigrade. Find the maximum temperature on this surface. Correct answer: 12. Explanation: Version 084 – Homework 12 – Gilbert – (59825) 2 We have to maximize the function T sub- ject to the constraint x 2 + y 2 + z 2 = 48 . The corresponding Lagrange function is F ( x, y, z, λ ) = x + y + z + λ ( x 2 + y 2 + z 2- 48) . At the critical point of F , therefore, F x = 1 + 2 λx = 0 , F y = 1 + 2 λy = 0 , F z = 1 + 2 λz = 0 , F λ = x 2 + y 2 + z 2- 48 = 0 . Thus x = y = z = radicalbig 48 / 3 = 4 ....
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## This note was uploaded on 05/08/2008 for the course M 408 M taught by Professor Gilbert during the Fall '07 term at University of Texas.

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HW_12_Key - Version 084 – Homework 12 – Gilbert...

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