To determine the feasible region for the production of Creamy Vanilla (x) and Continental Mocha (y), we start by establishing the constraints based on the ingredients available.

1. Egg Constraint:

   • Each quart of Creamy Vanilla requires 1 egg: $x$
   • Each quart of Continental Mocha requires 2 eggs: $2y$
   • Total eggs available: $800$ $x + 2y \leq 800$

2. Cream Constraint:

   • Each quart of Creamy Vanilla requires 2 cups of cream: $2x$
   • Each quart of Continental Mocha requires 2 cups of cream: $2y$
   • Total cream available: $1000$ $2x + 2y \leq 1000 \quad \text{(simplifies to)} \quad x + y \leq 500$

Now, we will find the intersection points of these constraints:

Finding Intersections

1. From $x + 2y = 800$:

   • If $x = 0$: $2y = 800 \Rightarrow y = 400 \quad (0, 400)$
   • If $y = 0$: $x = 800 \quad (800, 0)$

2. From $x + y = 500$:

   • If $x = 0$: $y = 500 \quad (0, 500)$
   • If $y = 0$: $x = 500 \quad (500, 0)$

Summary of Intersection Points

• $(0, 400)$ from the egg constraint
• $(800, 0)$ from the egg constraint
• $(0, 500)$ from the cream constraint
• $(500, 0)$ from the cream constraint

Feasible Region Coordinates

The feasible region is bounded by:

1. $(0, 0)$
2. $(0, 400)$
3. $(400, 0)$
4. $(500, 0)$

The feasible region will be where:

• $x \geq 0$
• $y \geq 0$

Coordinates of the vertices of the feasible region:

• $(0, 0)$
• $(0, 400)$
• $(500, 0)$
• $(400, 0)$

Would you like details on graphing this region or any specific questions? Here are some related questions for you:

1. How do you plot these constraints on a graph?
2. What is the significance of the vertices of the feasible region?
3. Can you determine the maximum production of each flavor?
4. How would changes in the availability of eggs or cream affect the feasible region?
5. What methods can be used to optimize production based on these constraints?

Tip: Always check the inequalities to ensure the correct feasible region is graphed!