15.8%20Ex%2033%20-%2034

15.8%20Ex%2033%20-%2034 - S E C T I O N 15.8 Lagrange...

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SECTION 15.8 Lagrange Multipliers: Optimizing with a Constraint (ET Section 14.8) 791 By the Length Formula we have AP = q ( x a ) 2 + ( y b ) 2 PB = q ( x c ) 2 + ( y d ) 2 The distance traveled by the runner is f ( x , y ) = q ( x a ) 2 + ( y b ) 2 + q ( x c ) 2 + ( y d ) 2 We must minimize the function f subject to the constraint g ( x , y ) = 0 (since the point P = ( x , y ) must satisfy the equation of the river). (b) The level curves of f ( x , y ) are f ( x , y ) = k for positive constants k .Thatis , q ( x a ) 2 + ( y b ) 2 + q ( x c ) 2 + ( y d ) 2 = k The level curve consists of all the points P = ( x , y ) such that the sum of the distances to the two ±xed points A = ( a , b ) and B = ( c , d ) is constant k > 0. Therefore the level curves are ellipses with foci at A and B . (c) The point P that minimizes the length of the path must satisfy the Lagrange Condition f P = λ g P ,the gradients of f and g are parallel vectors. Since the gradient at P is orthogonal to the level curve of the function passing through P , the level curve of f through P (which is the ellipse through P ) is tangent to the level curve of g through P , that is, it is tangent to the river. (d) The path-minimizing point P is the point such that the ellipse through P is tangent to the river. This point is shown in the ±gure below. River P A B g ( x , y ) = 0 33. Let L be the minimum length of a ladder that can reach over a fence of height h to a wall located a distance b behind the wall. (a) Use Lagrange multipliers to show that L = ( h 2 / 3 + b 2 / 3 ) 3 / 2 (Figure 16). Hint: Show that the problem amounts to minimizing f ( x , y ) = ( x + b ) 2 + ( y + h ) 2 subject to y / b = h / x or xy = bh .

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This homework help was uploaded on 04/22/2008 for the course MATH 32A taught by Professor Gangliu during the Winter '08 term at UCLA.

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15.8%20Ex%2033%20-%2034 - S E C T I O N 15.8 Lagrange...

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