Question 887173: A company manufactures bookshelves and desks for computers. The company’s profits are $25 per bookshelf and $55 per desk. To maintain high quality, the company should not manufacture more than a combined total of 80 bookshelves and desks per day. To meet customer demand, the company must manufacture between 30 and 80 bookshelves per day, inclusive. Furthermore, the company must manufacture at least 10 and no more than 30 desks per day. What is the maximum value for the objective function? What are the vertices for the feasible region.
Answer by Theo(13342)   (Show Source):
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the feasible region is above the line of y = 10 and below the line of y = 30 and to the right of the line x = 30 and to the left of the line x = 80 and below the line of y = 80 - x
the boundary points of the feasible region are (30,30) and (30,10) and (50,30) and (70,10).
your maximum profit will be at one of these boundary points.
your profit equation is p = 25*x + 55*y
x = number of bookshelves
y = number of desks
when x = 30 and y = 30, profit is 30*25 + 30*55 = 2400
when x = 30 and y = 10, profit is 30*25 + 10*55 = 1300
when x = 50 and y = 30, profit is 50*25 + 30*55 = 2900 *****
when x = 70 and y = 10, profit is 70*25 + 10*55 = 2300
maximum profit is when you make 50 bookshelves and 30 desks.
the line y = 80-x is derive from the equation x + y <= 80 which is the maximum of bookshelves or desks that can be made.
the feasible region satisfies all the constraints.
number of bookshelves and desks can't be greater than 80 (y <= 80-x).
number of bookshelves has to be greater than or equal to 30 and less than or equal to 80 (x >= 30, x <= 80).
the number of desks has to be greater than or equal to 10 and less than or equal to 30 (y>= 10, y <= 30).
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