265Lesson13Problems(3)

265Lesson13Problems(3) - cubic feet Determine the...

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MATH 265: MULTIVARIABLE CALCULUS: LESSON 13 PROBLEMS Problems are worth 20 points each. Warning: This assignment is going to take quite a bit of time! (1) Use the method of Lagrange Multipliers to determine the maximum and minimum of f ( x, y ) = 2 x - 3 y subject to the constraint g ( x, y ) = 4 - x 2 - 2 y 2 = 0. (2) Use the method of Lagrange Multipliers to determine the maximum f ( x, y ) = xy subject to the constraint g ( x, y ) = 6 x 2 + y 2 - 8 = 0. (3) Use the method of Lagrange Multipliers to determine the maximum and minimum of f ( x, y, z ) = x + y + z subject to the two conditions g ( x, y, z ) = x 2 + y 2 - 2 = 0 and h ( x, y, z ) = x + z - 1 = 0. (4) Use the method of Lagrange Multipliers to determine the maximum value of f ( x, y ) = x a y b (the a and b are two fixed positive constants) subject to the constraint x + y = 1. (Assume x, y > 0.) (5) Note: This is the same bonus problem as Lesson 12. This time, you have to use the method of Lagrange Multipliers to solve it. A rectangular tank with a bottom and sides but no top is to have volume 500
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Unformatted text preview: cubic feet. Determine the dimensions (length, width, height) with the smallest possible surface area. (6) (Bonus question: worth 10 points. Total points for assignment not to exceed 100.) A window is to be built in the shape of a rectangle surmounted by an isosceles triangle. The area of the window must be 6m 2 . Use Lagrange Multipliers to find the width and height of the rectangle for which the perimeter of the window will be as small as possible. (Clarifications: (1) A rectangle surmounted by a triangle has the shape of a stick drawing of a house as might be drawn by a child, a rectangle with a triangle on top. (2) The window’s perimeter is made up of three sides of the rectangle and the two equal length sides of the triangle. Draw a diagram! (3) This may require some industrial strength algebra!) 1...
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