Let $x$, $y$, and $z$ represent the number of packages Federal Rent-a-Car should buy from Fred Motors, Admiral Motors, and Chrysalis, respectively. The goal is to minimize the total cost, subject to the constraints that Federal needs at least 500 small cars, 400 medium cars, and 500 large cars.

Step 1: Define the Cost Function

The total cost of purchasing packages is: $\text{Cost} = 500,000x + 400,000y + 300,000z$

Step 2: Define the Constraints

Each manufacturer offers a different number of small, medium, and large cars in their packages. We can translate Federal’s minimum car requirements into the following constraints:

1. Small cars: $5x + 5y + 10z \geq 500$
2. Medium cars: $5x + 10y + 5z \geq 400$
3. Large cars: $10x + 5y + 5z \geq 500$
4. Non-negativity: $x, y, z \geq 0$

Step 3: Set Up the Linear Program

The problem can now be framed as a linear programming problem where we seek to:

Minimize

$500,000x + 400,000y + 300,000z$

Subject to

$5x + 5y + 10z \geq 500$ $5x + 10y + 5z \geq 400$ $10x + 5y + 5z \geq 500$ $x \geq 0, \quad y \geq 0, \quad z \geq 0$

Step 4: Solve the Linear Program

By solving this linear programming problem, we can determine the values of $x$, $y$, and $z$ that minimize the cost while meeting the car requirements.

If you'd like, I can solve this using computational tools to obtain the exact values for $x$, $y$, and $z$.

Would you like me to proceed with the calculations?

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Related Questions

1. How would changes in the minimum car requirements affect the solution?
2. What would happen if Federal could purchase packages in fractions?
3. How would adding a fourth manufacturer with different package offers change the solution?
4. What impact would an increase in the cost of each package have on the decision?
5. Can Federal save more by slightly increasing one of its minimum requirements?

Tip: In linear programming, constraints define feasible regions, which may be visualized graphically in simple problems, helping in understanding the optimal solution's location.