Question

2) (30) Solve questions below by Langrange method. a) A firm manufactures a good from two raw materials, X and Y. The quantity of its good which is produced from x units of X and y units of Y is given by Q(x,y) = x1/4y3/4 If the firm spends $1280 each week on the raw materials, what is its maximum possible weekly production, given that one unit of X costs $16 and one unit of Y costs $1? b) By maximizing and minimizing x2 + y2 subject to the constraint x2 + xy + y2 =3, Find the minimum and maximum distances from the origin to the ellipse with equation x2 + xy + y2 =3.
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Answer 1

ANSWER: A firm manufactures a good from two raw materials, X and y The quantities of its good which is produced from x units and X and Y Units Of Y is Q(2,4)= X x 44 y 3/4 also O (1) 16x+y = 1280 #16x+y-1280 By using Lagrange funcion, VR (20,42 avhay) 20 or 00 ah - :-( Monte) O2 80 00 de ya » Oh On 20 oy a oh ay xel 4314 2374 - XX16 g44 314 T 649 Уч 볼 w (**) 3 - 64% 216 25 14 => AS (393/4

į value in 3314 114 (8) im 4 lly hi ( १ ) ()" ģ. $ → y = 3x 4 Substitute y value in L1) 16x + 3x = 1280 X: 1280 19 3 x 1280 19 The Maximum pooduction is, 1280 114 Q 1280 ( 3* .) 34 max 19 bi Qmax 1280 19 x 3244 b) d (1,y) = x+42 C(x,y) = 422y+y<3=0 (D4 abumubuor hig multiplier method Chrono ax = chood pa OC ad वर 2x (22 , ay) (x+he h+77)

2x + 7 (24+24) 245 0124731) 291 24 - zaty dytx &x 2274 rey ayti axty ayta axy +22 day ty? x2 Ols =Y² ty x=y in (1) then X24022+R? -3=0 x=-y in o) 22.82702- 3 - 0 21 2: EV3 = (p1,-1) i. points are (1,1), 61,-1). (vo, -v3), (-13, 13) At ( 13,-13) and (3,18) dix,y) 343-6 (V3,-V3) and at distance is maxi (-V3, va) are points At (1,19 and 61,-1) dre y 124 12=2 (101) and (1,1) are are the points at ushen distance is moimum Thanking you please give like it helps to motivate me to do more wook friend.