Question

Comfort-by-Design Furniture produces chairs and sofas. Each chair requires 20 ft of wood, 1 lb of foam rubber, and 2 yd2 of fabric. Each sofa requires 100 ft of wood, 50 lb of foam rubber, and 20 yd2 of fabric. The manufacturer has in stock 1800 ft of wood, 500 lb of foam rubber, and 240 yd2 of fabric. The chairs can be sold for $180 each and the sofas for $1125 each. How many of each should be produced in order to maximize income? What is the maximum income? The manufacturer should make chairs and | sofas to yield a maximum income of $

Answer 1

solutions Given dath Materials Each Item Requires stock available clair Sofa wlood zoft 100 ft 1800.17 1 lb so 16 Foam Rubber soo lb 2 yd² 20 Yd fabric 240 ya? chair as $180 per chair 1125 persofa let c' be the number of chairs and 's' be the number of sofas manufactured. Therefore, we need to maximize the income Selling price S sofa 2 180 C +11255 Subject to : 202 + roos L 1800 Ctsos s soo 20+ 205 ? 240 C,S >0 Items can't be negative. C=0 =) S-10 S=0 ) Casoo from equation for, cao =) S=18. 2 cortinates S=0) C = 90 (Cs) Similarly from 1943 for, 0,18), (990) 3 0910 (500,0) from equy for o] (0,12), (120,0) Now plot these eqron Ø graph to get feasible Region C-0-) S=12 S=0 → C = 120

1920) (20,0 2C+ 205<240 Now, poond B can be obtained by solving error (0,18) 1912 CODY How check feasible region by equating each line with CH50 SL 500 204+1005 £1800) origin (0,0) 20 C+ 100S - 1800 for line 041800 b) It satisfies origin (0o) Therefore feasible region will be left sides of line. Similarly other or Two equation also satisfied origin (oo) Hence, our overall feasible region will be (o ABC) with origin at o (0,0) 13 206 + 100 S = 1800 C+ 50 S = 500

Multiply equ ② by 2 and subtract from egno re 20C +1005 = 1800 20+loos = 1000 1 8C to 800 --c= 44.44 C750S = 500 => 44.44750 S = Soo S = soo-44.44 SO S2 9.11 Therefor cordinate will be (44.44,9.11) Since (Cs) → Item (chair, sofa) i It should be integer je (44,9) Now, The maximum will be at the corners of OABC. 0100) => Max of 1800+ 1125 5 = A(0,10) >> Marx of B144,9) >> More of 18024 11255 = 180454 c (90,0) > Max. of 1800+ 11258 = Therefore Maximum income will be $ 18045 with uu chairs and a sofas. 180 et 1125 S = 11250 16200
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