Answer to Question #175739 in Statistics and Probability for kris

Question

1. An appliance dealer wants to purchase a combined total of no more than 100 refrigerators, and dishwashers for inventory. Refrigerators weigh 200 pound each, and dishwashers weigh 100 pounds each. The dealer is limited to a total of 12,000 pounds for these two items. A profit of $35 for each refrigerator and $20 on each dishwasher is projected.

(a) Write out the linear programming model by identifying the constraints and the objective function from the description above.

(b) Using a scale of 2 cm to 20 pounds on both axes, construct and shade the region R in which every point satisfies all the constraints.

(c) Based on the graph obtained in (b), determine the corner points and find out the maximum number of refrigerators and dishwashers that the dealer can purchase and sold to make the profit.

Answer 1

Let $x=$ the number of refrigerators, and $y=$ the number of dishwashers

$$x\geq 0, y\geq 0$$

An appliance dealer wants to purchase a combined total of no more than 100 refrigerators, and dishwashers for inventory

$$x+y\leq100$$

The dealer is limited to a total of 12,000 pounds for these two items

$$200x+100y\leq 12000$$

The total profit is

$$z=35x+20y$$

(a) linear programming problem maximizing $z=35x+20y$ subject to

$$x+y\leq100$$

$$200x+100y\leq 12000$$

$$x\geq 0, y\geq 0$$

(b)

(c)

Corner points: $O(0, 0), A(0, 100), B(20, 80), C(60, 0)$

$OA: x=0, 0\leq y\leq 100$

$$z=20y$$ $$0\leq z\leq 2000$$

$AB: x+y=100, 0\leq x\leq 20$

$$z=35x+20(100-x)=2000+15x$$ $$2000\leq z\leq 2300$$

$BC: 2x+y=120, 20\leq x\leq 60$

$$z=35x+20(120-2x)=2400-5x$$ $$2100\leq z\leq 2300$$

$CO: y=0, 0\leq x\leq 60$

$$z=35x$$ $$0\leq z\leq 2100$$

$B(20, 80): z=2300$

An appliance dealer should purchase 20 refrigerators, and 80 dishwashers in order to maximize his profit.

The maximum profit is $2300.