a paper manufacturing company recycles paper, cans and other sheet metal. the profit on the paper is $500 and the profit on the cans is $350 per pound
A) write the objective function that models the daily profit
B) the manufacture is bound by the following constraints , namely:
*equipment allows at most 200lbs of can a day at the same time
*equipment allows at most 200lbs of paper per day
*equipment require at least 10kbs of paper and 80lbs of cans to operate
*equipment allows for recycling at msot 300bs of cans per day
C) graph the inequalities from equations in B
D)evaluate the objective function at each of the vertices and determine the mix of cans and paper for max profit
x = paper
y = cans
p = 500 x + 350 y
y </= 200
x </= 200
x >/= 10
y >/= 80
I do not understand last constraint, redundant if y</=200
rectangle
corners at
(10, 80) (10,200) (200, 80) (200,200)
well, go ahead and evaluate p at each corner but obviously mas at (200,200)
A) To write the objective function that models the daily profit, let's define the variables:
x = pounds of paper recycled per day
y = pounds of cans and sheet metal recycled per day
The profit on paper is $500, and the profit on cans is $350 per pound. Therefore, the objective function can be written as:
Profit = 500x + 350y
B) Now let's consider the constraints given:
1) The equipment allows at most 200lbs of cans per day:
y ≤ 200
2) The equipment allows at most 200lbs of paper per day:
x ≤ 200
3) The equipment requires at least 10lbs of paper and 80lbs of cans to operate:
x ≥ 10
y ≥ 80
4) The equipment allows for recycling at most 300lbs of cans per day:
y ≤ 300
C) To graph the inequalities, we can plot them on a coordinate plane.
First, plot the line y = 200, which represents the constraint y ≤ 200. This line will be horizontal and pass through the point (0, 200). Shade the region below the line, as the inequality is y ≤ 200.
Next, plot the line x = 200, which represents the constraint x ≤ 200. This line will be vertical and pass through the point (200, 0). Shade the region to the left of the line, as the inequality is x ≤ 200.
Now, plot the line y = 80, which represents the constraint y ≥ 80. This line will be horizontal and pass through the point (0, 80). Shade the region above the line, as the inequality is y ≥ 80.
Lastly, plot the line y = 300, which represents the constraint y ≤ 300. This line will be horizontal and pass through the point (0, 300). Shade the region below the line, as the inequality is y ≤ 300.
The shaded region where all the shaded areas overlap represents the feasible region.
D) To evaluate the objective function at each of the vertices, identify the points where the boundary lines intersect:
1) (10, 80)
2) (10, 200)
3) (200, 80)
4) (200, 200)
Now substitute these values into the objective function:
1) Profit = 500(10) + 350(80) = $35,000
2) Profit = 500(10) + 350(200) = $87,000
3) Profit = 500(200) + 350(80) = $160,000
4) Profit = 500(200) + 350(200) = $110,000
From these results, we can see that the maximum profit is obtained at the vertex (200, 80), which means recycling 200lbs of paper and 80lbs of cans per day.