give answer in 5 step with explanation at the end of each step and final answer at the end:
Some factory stocks
4
5
kg of material A and
2
7
kg of material B
.
In order to
1
L of good X
,
we use
5
kg of A and
2
kg of B
.
In order to
1
L of good Y
,
we use
3
kg of A and
3
kg of B
.
The profits are
1
2
thousands JPY
/
1
L of X and
1
1
thousands JPY
/
1
L of Y
.
Then, how much do you make X and Y in order to get the maximum profit
?
More, how much the maximum profit
?Some factory stocks 45kg of material A and 27
kg of material B. In order to 1L of good X, we
use 5kg of A and 2kg of B. In order to 1L of
good Y, we use 3kg of A and 3kg of B. The
profits are 12 thousands JPY/1L of X and 11
thousands JPY/1L of Y. Then, how much do
you make X and Y in order to get the
maximum profit ? More, how much the
maximum profit ?
Question:
give answer in 5 step with explanation at the end of each step and final answer at the end:
Some factory stocks
4
5
kg of material A and
2
7
kg of material B
.
In order to
1
L of good X
,
we use
5
kg of A and
2
kg of B
.
In order to
1
L of good Y
,
we use
3
kg of A and
3
kg of B
.
The profits are
1
2
thousands JPY
/
1
L of X and
1
1
thousands JPY
/
1
L of Y
.
Then, how much do you make X and Y in order to get the maximum profit
?
More, how much the maximum profit
?Some factory stocks 45kg of material A and 27
kg of material B. In order to 1L of good X, we
use 5kg of A and 2kg of B. In order to 1L of
good Y, we use 3kg of A and 3kg of B. The
profits are 12 thousands JPY/1L of X and 11
thousands JPY/1L of Y. Then, how much do
you make X and Y in order to get the
maximum profit ? More, how much the
maximum profit ?
This is AI generated Response.
Asked by: saravjeet
Created at: 2025-05-27 21:21:34
Final Answer:
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Let's solve the problem step by step:
---
### **Step 1: Define Variables**
Let
- \( x \) = liters of X produced
- \( y \) = liters of Y produced
**Explanation:**
We need to decide how much of X and Y to produce, so we assign variables to these quantities.
---
### **Step 2: Formulate Constraints**
**Material A constraint:**
Each X needs 5 kg A, each Y needs 3 kg A, total available is 45 kg:
\[
5x + 3y \leq 45
\]
**Material B constraint:**
Each X needs 2 kg B, each Y needs 3 kg B, total available is 27 kg:
\[
2x + 3y \leq 27
\]
**Non-negativity constraints:**
\[
x \geq 0,\ y \geq 0
\]
**Explanation:**
We use the material requirements and availability to set up inequalities.
---
### **Step 3: Write Objective Function**
Profit per X: 12 thousand JPY
Profit per Y: 11 thousand JPY
So, maximize
\[
P = 12x + 11y
\]
**Explanation:**
We write the equation to maximize based on profit per product.
---
### **Step 4: Find the Corner Points**
Solve the system for intersection points:
**a. \( x=0 \):**
\[
3y \leq 45 \implies y \leq 15\\
3y \leq 27 \implies y \leq 9
\]
So, at \( x=0 \), \( y=9 \) is feasible.
**b. \( y=0 \):**
\[
5x \leq 45 \implies x \leq 9\\
2x \leq 27 \implies x \leq 13.5
\]
So, at \( y=0 \), \( x=9 \) is feasible.
**c. Intersection of the two constraints:**
\[
5x + 3y = 45\\
2x + 3y = 27
\]
Subtract the second from the first:
\[
(5x + 3y) - (2x + 3y) = 45 - 27\\
3x = 18\\
x = 6
\]
Plug \( x=6 \) into \( 5x+3y=45 \):
\[
5(6) + 3y = 45\\
30 + 3y = 45\\
3y = 15\\
y = 5
\]
So, intersection point is \( (6,5) \).
**Explanation:**
We find the feasible region's corner points (where constraints meet).
---
### **Step 5: Evaluate Objective Function at Corner Points**
- At \( (x, y) = (0, 9) \): \( P = 12*0 + 11*9 = 99 \)
- At \( (x, y) = (9, 0) \): \( P = 12*9 + 11*0 = 108 \)
- At \( (x, y) = (6, 5) \): \( P = 12*6 + 11*5 = 72 + 55 = 127 \)
**Explanation:**
We check profit at each feasible corner point to find the **maximum**.
---
## **Final Answer**
**To maximize profit, produce 6 liters of X and 5 liters of Y.
The maximum profit is 127 thousand JPY.**
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Some factory stocks 45kg of material A and 27
kg of material B. In order to 1L of good X, we
use 5kg of A and 2kg of B. In order to 1L of
good Y, we use 3kg of A and 3kg of B. The
profits are 12 thousands JPY/1L of X and 11
thousands JPY/1L of Y. Then, how much do
you make X and Y in order to get the
maximum profit ? More, how much the
maximum profit ?