Question

A company can use cotton to produce gloves, hats, and shirts. The profit on a pair of gloves(G) is $5, the profit on a hat(H) is $9, and the profit on a shirt(s) is $10. The items are produced according to the following production functions: G-36CG H = 23CH-0.5CH? 5 - 80C5-C2 where CG. CH. and Cs are the pounds of cotton There are 120 pounds of cotton available. How many pounds of cotton should be used to make hatss (CH)? lon Enter as a value.

Answer 1

Total profit function is a s follows: Profit = P&+PH+PS = 5G +9H+10S = 5(3606)+9(23CH – 0.5C7°) +10 (80C5-Cg2) = 180C6 +207C4 - 4.5C ? +800C5 -10C2 The profit maximization constraint is as follows: Cc+CH +Cs = 120 Set up the lagrangian function as follows: L=Profit +1(120-C6-CH-Cs) It can be written as: L=180C6 +207C7 - 4.5C, +800C, -10C, +1(120-C6-CH-Cs) Calculate the value of 2 as follows:

al OL = 180-2 ac 0 = 180-2 2 = 180 Thus, a = 180 Calculate the pounds of cotton that should be used to make hats (CH) as follows: OCH PL = (180C, +207C7 - 4.5C? +800C, – 100,2 +2(120-Co-C7-C3)) 207-9C -1 = 207-9C-180 9C4 = 27 Ct = 27 Thus, the pounds of cotton that should be used to make hats(CH) is 3 pounds. Calculate the pounds of cotton that should be used to make shirts (Cs) as follows:

* = (180Cc +207C7 - 4.5C,? + 800C5 – 100g? +2(120-Co-Cy -Cs)) = 800 – 2005 - 2 = 620 – 2005 2005 = 620 Cs= 20 = 31 Thus, the pounds of cotton that should be used to make shirts (Cs) is 31 pounds). Calculate the pounds of cotton that should be used to make gloves (CG) as follows: al OCX 4 =120-CG -CH-Cs 0 = 120-C6-3-31 Co = 120-34 = 86 Thus, the pounds of cotton that should be used to make gl ounds Thus, the pounds of cotton that should be used to make l ounds