# HW 8 - Math E-21a Fall 2009 HW#8 problems Read sections...

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1 Math E-21a – Fall 2009 – HW #8 problems Read sections 11.8 and 12.1 (and maybe 12.2-12.3) and do the following problems: Problems to turn in on Thurs, Oct 29 : 1. (Prob. 11.8/16) Use the Method of Lagrange Multipliers to find the maximum and minimum values of the function (, ,) 3 3 f xyz x y z  subject to the two constraints 0 xyz  and 22 21 xz  . 2. (Prob. 11.8/18) Find the extreme values of the function (, ) 2 3 4 5 f xy x y x  in the region described by the inequality 16 xy  . 3. (Probs. 11.8/21-22) a) The total production P of a certain product depends on the amount L of labor used and the amount K of capital investment. In Sections 11.1 and 11.3 we discussed how the Cobb-Douglass model 1 Pb LK follows from certain economic assumptions, where b and are positive constants and 1 . If the cost of a unit of labor is m and the cost of a unit of capital is n , and the company can spend only p dollars as its total budget, then maximizing the production P is subject to the constraint mL nK p . Show that the maximum production occurs when p L m and (1 ) p K n . b) If we now assume that the production is fixed at 1 bL K Q  , where Q is a constant, what values of L and K will minimize the cost function (, ) CLK mL nK ? 4. (Prob. 11.8/24) Use the Method of Lagrange Multipliers to prove that the triangle with maximum area that has a given perimeter p is equilateral. [ Hint : Use Heron’s formula for the area: () A ss x s y s z , where 2 p s and x , y , and z are the lengths of the sides of the triangle.] 5. (Prob. 11.8/38) Find the maximum and minimum volumes of a rectangular box whose surface area is 1500 cm 2 and whose total edge length is 200 cm. [ Note : Due to the multiple constraints, it’s not hard to show that in neither case will the box be a cube!] 6. (Prob. 11.8/40) The plane 4 3 8 5  intersects the cone 222 zxy in an ellipse.

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HW 8 - Math E-21a Fall 2009 HW#8 problems Read sections...

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