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A paper recycling company uses scrap cloth and scrap paper to make two different grades of recycled paper. A single batch of grade A recycled paper is made from 25 lb of scrap cloth and 10 lb of scrap paper, whereas one batch of grade B recycled paper is made from 10 lb of scrap cloth and 20 lb of scrap paper. The company has 100 lb of scrap cloth and 120 lb of scrap paper on hand. A batch of grade A paper brings a profit of $500, whereas a batch of grade B paper brings a profit of $250. What amounts of each grade should be made? How, if at all, do the maximum profit and optimal production policy change if the company is required to produce at least one batch of each type?

Question

A paper recycling company uses scrap cloth and scrap paper to make two different grades of recycled paper. A single batch of grade A recycled paper is made from 25 lb of scrap cloth and 10 lb of scrap paper, whereas one batch of grade B recycled paper is made from 10 lb of scrap cloth and 20 lb of scrap paper. The company has 100 lb of scrap cloth and 120 lb of scrap paper on hand. A batch of grade A paper brings a profit of $500, whereas a batch of grade B paper brings a profit of $250. What amounts of each grade should be made? How, if at all, do the maximum profit and optimal production policy change if the company is required to produce at least one batch of each type?

check_circleAnswer
Step 1

Two different grades of recycled paper are given.

The details can be tabulated as follows:
Grade A
Grade B
Total
Cloth
25
10
100
120
Раper
10
20
Profit
500
250
Let the amount of grade A paperbe s and the amount of grade B paperbet
w
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The details can be tabulated as follows: Grade A Grade B Total Cloth 25 10 100 120 Раper 10 20 Profit 500 250 Let the amount of grade A paperbe s and the amount of grade B paperbet w

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Step 2

The constraints can be built as follows.

25s 10t100
10s 20t 120
The maximized function becomes P = 500s +250t
The s-intercept for 25s +10t 100 is 4.
The s-intercept for 10s +20t 120 is 12.
The t-intercept for 25s 10t 100 is 10.
The f-intercept for 10s 20t 120 is 6
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Image Transcriptionclose

25s 10t100 10s 20t 120 The maximized function becomes P = 500s +250t The s-intercept for 25s +10t 100 is 4. The s-intercept for 10s +20t 120 is 12. The t-intercept for 25s 10t 100 is 10. The f-intercept for 10s 20t 120 is 6

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Step 3

Solve the related equat...

25s 10t 100
5s 2t 20
10s20t 1202s4t24
5s 2t 20
x2
2s4 24 x5
10s 4t 40
10s20t 120
-16t 80t = 5
Then, s 2
The amount of grade A paper is 2 and the amount of grade B paper is 5
help_outline

Image Transcriptionclose

25s 10t 100 5s 2t 20 10s20t 1202s4t24 5s 2t 20 x2 2s4 24 x5 10s 4t 40 10s20t 120 -16t 80t = 5 Then, s 2 The amount of grade A paper is 2 and the amount of grade B paper is 5

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