1. **State the problem:** The Miers Company produces engines to be shipped to two plants, Plant I and Plant II. Plant I requires at least 45 engines, Plant II requires at least 32 engines, and the total shipment to both plants cannot exceed 120 engines. The shipping cost is $30 per engine to Plant I and $40 per engine to Plant II. The goal is to analyze the shipping quantities and cost constraints. 2. **Define variables:** Let $x$ be the number of engines shipped to Plant I and $y$ be the number shipped to Plant II. 3. **Write inequalities based on requirements:** - Plant I minimum: $x \geq 45$ - Plant II minimum: $y \geq 32$ - Total shipment limit: $x + y \leq 120$ 4. **Cost function:** The total shipping cost $C$ is $$C = 30x + 40y$$ 5. **Interpretation:** To minimize cost, the company wants to find $x$ and $y$ satisfying the restrictions while minimizing $C$. 6. **Summary:** The problem involves linear inequalities for shipping quantities and a linear cost function. Final answer: Shipping quantities must satisfy $$x \geq 45,\quad y \geq 32,\quad x + y \leq 120$$ with total cost $$C = 30x + 40y.$$