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The problem provided describes a manufacturing company receiving orders for engines from two assembly plants, with conditions about the number of engines needed, shipping costs, and rebates.

Let's break down the given information step by step:

1. **Engine Requirements**:
   - Plant I needs **at least 45 engines**.
   - Plant II needs **at least 32 engines**.
   - The company can send a maximum of **140 engines**.

2. **Costs and Rebates**:
   - It costs **$30 per engine** to ship to **Plant I**.
   - It costs **$50 per engine** to ship to **Plant II**.
   - Plant I gives a rebate of **$20 per engine**.
   - Plant II gives a rebate of **$15 per engine**.
   - The manufacturer estimates that they need at least **$1500 in rebates** to cover products.

3. **Goal**:
   - We need to determine how many engines should be shipped to each plant to minimize shipping costs while satisfying all conditions.

### Variables:
Let:
- $ x $ = number of engines shipped to Plant I.
- $ y $ = number of engines shipped to Plant II.

### Constraints:
1. $ x + y \leq 140 $ (the total number of engines shipped cannot exceed 140).
2. $ x \geq 45 $ (Plant I needs at least 45 engines).
3. $ y \geq 32 $ (Plant II needs at least 32 engines).
4. $ 20x + 15y \geq 1500 $ (the total rebate must be at least $1500).

### Objective:
Minimize the shipping cost: 
\[
30x + 50y
\]

This is a linear programming problem, where we aim to minimize the shipping cost while satisfying all constraints. I'll calculate the optimal number of engines to ship to each plant using these conditions.To minimize the shipping cost while satisfying all the given constraints, the optimal solution is:

- **Ship 51 engines to Plant I**.
- **Ship 32 engines to Plant II**.

The **minimum shipping cost** is **$3,130**.

Would you like more details on how the solution was derived or have any other questions?

Here are 5 related questions you might consider:
1. How does the rebate affect the overall cost calculation?
2. What happens if we change the maximum number of engines to 150?
3. What is the impact of increasing the shipping cost for one of the plants?
4. How would the solution change if Plant I required fewer engines?
5. Can we use this method to optimize other types of production costs?

**Tip**: When dealing with constraints in optimization problems, always make sure the constraints are properly translated into inequalities or equalities for better accuracy.

A manufacturing company receives orders for engines from two assembly plants. Plant I needs at least 45 engines, and Plant II needs at least 32 engines. The company can send at most 140 engines to these assembly plants. It costs $30 per engine to ship to Plant I and $50 per engine to ship to Plant II. Plant I gives the manufacturing company $20 in rebates toward its products for each engine they buy, while Plant II gives similar $15 rebates. The manufacturer estimates that they need at least $1500 in rebates to cover products they buy from the two plants. How many engines should be shipped to each plant to minimize shipping costs? What is the minimum cost?

This problem involves optimizing shipping costs for a manufacturing company sending engines to two assembly plants with rebates and shipping costs. Learn how to use linear programming to determine the minimum shipping cost while meeting specific engine and rebate constraints.

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