1. a. Constraints:            6x + 3y < 50
                 	     3x + 8y < 60
	                      4x + 2y < 35
	                      x > 0, y > 0

  			   profit: P = 4x + 6y

   b. 
i. 6∙5 + 3∙5 < 50 is true; 3∙5 + 8∙5 < 60 is true; 4∙5 + 2∙5 < 35 is true; so this production       policy is feasible. Profit would be $50
	
        ii. 6∙5 + 3∙7 < 50 is not true, so the production policy is not feasible.

2. corner points: intersection of 2x + 3y = 26 and x = 1: (1, 8);
               intersection of 4x + 4y = 40 and y = 2: (8, 2);
               intersection of 2x + 3y = 26 and 4x + 4y = 40: (4, 6)

feasible region:

corner pnt	profit (5x + 3y)	
 (1, 8)	5 + 24 = 29	
  (8, 2)	40 + 6 = 46	
(4, 6)	20 + 18=38	
Solution: maximum occurs when x = 8, y = 2; maximum value is 46

3. 
Let x and y be the amounts of grade A and grade B recycled paper that the company should make. Note that x and y must be grater than or equal to 0. 
To produce these amounts, the company needs to use:  lb of scrap cloth and lb of scrap paper. Therefore, the constraints are: ; ;  and . 
The total profit of x batches of grade A paper and y batches of grade B paper is: 

4. 
The first step is finding the feasible
region from the given constraints by 
sketching the inequalities. 
This feasible region has four corners:

*The first corner:

*The second corner: 

*The third corner: 

*The fourth corner: 

Corner	P = 10x + 100y	
(0,0)	P = 0	
(0,4)	P = 400	
(6,0)	P = 60	
(5,2)	P = 250	

Plugging the values of these corners into the profit function
gives us: the maximum value of P is 400 when x = 0 
and y = 4.