ChemLabs uses raw materials / and // to produce two domestic cleaning solutions, A and B. The daily availabilities of raw materials / and // are 150 and 145 units, respectively. One unit of solution A consumes .5 unit of raw material / and .6 unit of raw material II, and one unit of solution B uses .5 unit of raw material / and .4 unit of raw material //. The profits per unit of solutions A and B are $8 and $10, respectively. The daily demand for solution A lies between 30 and 150 units, and that for solution B between 40 and 200 units. Find the optimal production amounts of A and B.

To solve this optimization problem, we need to maximize the profit while considering the constraints on raw materials and demand. This is a linear programming (LP) problem, where:

- **Objective:** Maximize profit by choosing optimal amounts of solutions $ A $ and $ B $ to produce.
- **Constraints:** Availability of raw materials, demand for solutions $ A $ and $ B $, and non-negativity of production amounts.

### Step 1: Define the decision variables
Let:
- $ x_1 $ = number of units of solution $ A $ produced.
- $ x_2 $ = number of units of solution $ B $ produced.

### Step 2: Write the objective function
The total profit $ P $ is given by:
\[
P = 8x_1 + 10x_2
\]
This is what we want to **maximize**.

### Step 3: Define the constraints
#### Raw material availability constraints:
- For raw material $ / $: 
  \[
  0.5x_1 + 0.5x_2 \leq 150
  \]
- For raw material $ // $: 
  \[
  0.6x_1 + 0.4x_2 \leq 145
  \]

#### Demand constraints:
- The demand for solution $ A $ lies between 30 and 150 units:
  \[
  30 \leq x_1 \leq 150
  \]
- The demand for solution $ B $ lies between 40 and 200 units:
  \[
  40 \leq x_2 \leq 200
  \]

#### Non-negativity constraints:
\[
x_1 \geq 0, \quad x_2 \geq 0
\]

### Step 4: Linear programming formulation
The problem can now be formulated as follows:

**Maximize:**
\[
P = 8x_1 + 10x_2
\]

**Subject to:**
1. $ 0.5x_1 + 0.5x_2 \leq 150 $ (raw material / constraint)
2. $ 0.6x_1 + 0.4x_2 \leq 145 $ (raw material // constraint)
3. $ 30 \leq x_1 \leq 150 $ (demand for A)
4. $ 40 \leq x_2 \leq 200 $ (demand for B)
5. $ x_1 \geq 0 $, $ x_2 \geq 0 $ (non-negativity constraints)

### Step 5: Solve using linear programming
To find the optimal solution, we can use a method such as the **Simplex Method** or a solver for linear programming. I will now calculate the optimal solution for this problem.The optimal production amounts for ChemLabs are:

- Produce **100 units** of solution $ A $.
- Produce **200 units** of solution $ B $.

This results in a **maximum profit of $2800** per day.

Would you like further details or have any other questions? Here are some related questions to explore further:

1. How would the solution change if the availability of raw material $ // $ increased to 160 units?
2. What happens to the profit if the profit per unit of solution $ B $ decreases to $9?
3. Can you explain the Simplex Method used in linear programming?
4. What if the demand for solution $ B $ decreases to a range of 30-100 units?
5. How would the solution be affected if both raw materials were more expensive?

**Tip:** Linear programming is useful for optimizing problems with constraints; graphical methods work for two variables, but solvers like Simplex handle larger cases efficiently.

ChemLabs uses raw materials / and // to produce two domestic cleaning solutions, A and B. The daily availabilities of raw materials / and // are 150 and 145 units, respectively. One unit of solution A consumes 0.5 unit of raw material / and 0.6 unit of raw material //, and one unit of solution B uses 0.5 unit of raw material / and 0.4 unit of raw material //. The profits per unit of solutions A and B are $8 and $10, respectively. The daily demand for solution A lies between 30 and 150 units, and that for solution B between 40 and 200 units. Find the optimal production amounts of A and B.

Discover how to optimize the production of cleaning solutions A and B at ChemLabs using linear programming. This problem involves maximizing profit under raw material constraints and demand limits, and is solved using methods like the Simplex Method. Perfect for advanced high school or undergraduate students.

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