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 ​$40 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 ​$10 rebates. The manufacturer estimates that they need at least ​$1500 in rebates to cover products they plan to 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 can be solved using linear programming, where we aim to minimize shipping costs subject to several constraints.

### Step 1: Define the Variables
Let:
- $ x_1 $ = number of engines shipped to Plant I
- $ x_2 $ = number of engines shipped to Plant II

### Step 2: Objective Function
The objective is to minimize the total shipping cost. The shipping cost for Plant I is $30 per engine and for Plant II is $40 per engine. Thus, the total shipping cost is:

\[
\text{Minimize } Z = 30x_1 + 40x_2
\]

### Step 3: Constraints
We have the following constraints:
1. **Plant I requires at least 45 engines**:
   \[
   x_1 \geq 45
   \]
2. **Plant II requires at least 32 engines**:
   \[
   x_2 \geq 32
   \]
3. **Total engines sent to both plants cannot exceed 140**:
   \[
   x_1 + x_2 \leq 140
   \]
4. **The manufacturer needs at least $1500 in rebates**:
   - Plant I provides a rebate of $20 per engine, so the total rebate from Plant I is $ 20x_1 $.
   - Plant II provides a rebate of $10 per engine, so the total rebate from Plant II is $ 10x_2 $.
   The total rebate must be at least $1500:
   \[
   20x_1 + 10x_2 \geq 1500
   \]

### Step 4: Set Up the Linear Programming Model
We now have the linear programming model:

\[
\text{Minimize } Z = 30x_1 + 40x_2
\]
subject to:
\[
x_1 \geq 45
\]
\[
x_2 \geq 32
\]
\[
x_1 + x_2 \leq 140
\]
\[
20x_1 + 10x_2 \geq 1500
\]

### Step 5: Solve the Problem
This problem can be solved graphically or using linear programming methods like the Simplex method. I'll calculate the solution for you using the conditions outlined.

Let me solve this.The optimal solution is to ship:
- $ x_1 = 59 $ engines to Plant I
- $ x_2 = 32 $ engines to Plant II

The minimum shipping cost is $3050.

Would you like further details or have any questions?

Here are 5 related questions:
1. What would happen if the rebate requirement increased to $2000?
2. How would the solution change if Plant II required at least 40 engines instead of 32?
3. Could we minimize costs further by relaxing the rebate condition?
4. What would the solution be if shipping costs were $25 per engine for Plant I?
5. How does the constraint on total engines affect the final solution?

**Tip**: In linear programming, constraints heavily influence the feasibility of the solution. Reducing the total allowed shipments or increasing rebate requirements can dramatically alter outcomes.

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 $40 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 $10 rebates. The manufacturer estimates that they need at least $1500 in rebates to cover products they plan to buy from the two plants. How many engines should be shipped to each plant to minimize shipping costs? What is the minimum cost?

Solve a linear programming problem where a manufacturing company must minimize shipping costs while meeting engine demand and rebate requirements. This problem involves constraints on engine shipments to two plants and rebate conditions, suitable for Grades 10-12 students.

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