A company makes two types of sofas, regular and long, at two locations, one in Hickory and one in Lenoir.

The plant in Hickory has a daily operating budget of $27000 and can produce at most 180 sofas daily in any combination. It costs $150 to make a regular sofa and $200 to make a long sofa at the Hickory plant. The Lenoir plant has a daily operating budget of $21600, can produce at most 150 sofas daily in any combination and makes a regular sofa for $135 and a long sofa for $180. The company wants to limit production to a maximum of 150 regular sofas and 210 long sofas each day.

The company makes a profit of $55 on each regular sofa and $77 on each long sofa.

How many of each type should be made at each plant in order to maximize profit?
constraints: x3+x4<=150, 150x1+200x2<=27000, x2+x4<=210
Write all of the constraints and the profit function using the following variable names:
x1= # regular sofas made at Hickory
x2= # long sofas made at Hickory
x3= # regular sofas made at Lenoir
x4= # long sofas made at Lenoir Solve the problem using the simplex method and answer the following questions. Do not use commas in your answers or they will be graded as incorrect.
In the optimal solution, how many regular sofas are produced?
 


In the optimal solution, how many long sofas are produced?
 


In the optimal solution, how many long sofas are produced in Hickory?


In the optimal solution, how many regular sofas are produced in Lenoir?


What is the maximum profit?
$ 


The problem you're dealing with is a linear programming problem where we need to maximize the profit function while satisfying the constraints related to production limits and budget at two different plants.

### Variables:
- $ x_1 $: Number of regular sofas made at Hickory
- $ x_2 $: Number of long sofas made at Hickory
- $ x_3 $: Number of regular sofas made at Lenoir
- $ x_4 $: Number of long sofas made at Lenoir

### Objective:
Maximize the profit function:
\[
\text{Profit} = 55x_1 + 77x_2 + 55x_3 + 77x_4
\]

### Constraints:
1. **Production limits at Hickory:**
   - $ 150x_1 + 200x_2 \leq 27000 $ (Hickory budget constraint)
   - $ x_1 + x_2 \leq 180 $ (Hickory production capacity)
   
2. **Production limits at Lenoir:**
   - $ 135x_3 + 180x_4 \leq 21600 $ (Lenoir budget constraint)
   - $ x_3 + x_4 \leq 150 $ (Lenoir production capacity)
   
3. **Overall sofa production limits:**
   - $ x_1 + x_3 \leq 150 $ (Total regular sofas must be at most 150)
   - $ x_2 + x_4 \leq 210 $ (Total long sofas must be at most 210)

4. **Non-negativity constraints:**  
   $ x_1, x_2, x_3, x_4 \geq 0 $

Now, let's solve this using the simplex method to find the optimal solution.

### Solution:

1. **Optimal number of regular sofas produced (total):**
2. **Optimal number of long sofas produced (total):**
3. **Optimal number of long sofas produced in Hickory:**
4. **Optimal number of regular sofas produced in Lenoir:**
5. **Maximum profit:**

I'll now calculate this.Here are the results of the optimization:

1. **Total number of regular sofas produced:** 60
2. **Total number of long sofas produced:** 210
3. **Number of long sofas produced in Hickory:** 90
4. **Number of regular sofas produced in Lenoir:** 0
5. **Maximum profit:** $19,470

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

1. How does the budget constraint affect production at both plants?
2. What would happen if the profit on regular sofas increased?
3. How much more profit could the company make if the production capacity at Lenoir increased?
4. What if the company wanted to produce more regular sofas?
5. How would reducing the budget at the Hickory plant affect overall profit?

**Tip:** Linear programming is a powerful tool to balance multiple constraints while optimizing outcomes, often used in production planning, finance, and logistics.

A company makes two types of sofas, regular and long, at two locations, one in Hickory and one in Lenoir. The plant in Hickory has a daily operating budget of $27000 and can produce at most 180 sofas daily in any combination. It costs $150 to make a regular sofa and $200 to make a long sofa at the Hickory plant. The Lenoir plant has a daily operating budget of $21600, can produce at most 150 sofas daily in any combination and makes a regular sofa for $135 and a long sofa for $180. The company wants to limit production to a maximum of 150 regular sofas and 210 long sofas each day. The company makes a profit of $55 on each regular sofa and $77 on each long sofa. How many of each type should be made at each plant in order to maximize profit?

This linear programming problem involves maximizing the profit of a company producing regular and long sofas at two locations, Hickory and Lenoir. The solution is found using the simplex method, taking into account production budgets, capacity, and profit margins. Learn how to optimize production to achieve maximum profit. Suitable for advanced high school or college-level students.

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