Part (a) What can the manager ask for directly?

The manager can directly ask for:

• The number of operations at Refinery 1 (denoted by $x$).
• The number of operations at Refinery 2 (denoted by $y$).

These numbers will then determine how much of each grade of petroleum (AA, BB, and CC) is produced.

Part (b) Total amount of AA produced

Each operation at Refinery 1 produces 2 units of AA, and each operation at Refinery 2 produces 5 units of AA. So, the total amount of AA produced is given by:

$AA(x, y) = 2x + 5y$

Part (c) Total amount of BB produced

Each operation at Refinery 1 produces 1 unit of BB, and each operation at Refinery 2 produces 1 unit of BB. Therefore, the total amount of BB produced is:

$BB(x, y) = x + y$

Part (d) Total amount of CC produced

Each operation at Refinery 1 produces 5 units of CC, and each operation at Refinery 2 produces 2 units of CC. Hence, the total amount of CC produced is:

$CC(x, y) = 5x + 2y$

Part (e) Inequalities for each grade of petroleum

The consumer needs at least:

• 540 units of AA,
• 180 units of BB,
• 420 units of CC.

This gives us the following system of inequalities based on the production from $x$ operations at Refinery 1 and $y$ operations at Refinery 2:

1. For AA: $2x + 5y \geq 540$
2. For BB: $x + y \geq 180$
3. For CC: $5x + 2y \geq 420$

Finding the feasible region

To solve this graphically, we treat the inequalities as equations and plot them on the $xy$-plane:

1. For $2x + 5y = 540$, the intercepts are $(270, 0)$ and $(0, 108)$.
2. For $x + y = 180$, the intercepts are $(180, 0)$ and $(0, 180)$.
3. For $5x + 2y = 420$, the intercepts are $(84, 0)$ and $(0, 210)$.

Next, we solve the system to find the corners (vertices) of the feasible region by finding the intersection points of these lines.

1. Intersection of $2x + 5y = 540$ and $x + y = 180$: Solving these equations, we get $x = 150$ and $y = 30$. $(150, 30)$

2. Intersection of $2x + 5y = 540$ and $5x + 2y = 420$: Solving these, we get $x = 180$ and $y = 36$. $(180, 36)$

3. Intersection of $x + y = 180$ and $5x + 2y = 420$: Solving these, we get $x = 120$ and $y = 60$. $(120, 60)$

4. Finally, the intercept $(270, 0)$ on the $2x + 5y = 540$ line also serves as a boundary point.

Feasible basic solutions (in increasing order of $x$-coordinate):

The feasible basic solutions are:

$(120, 60), (150, 30), (180, 36), (270, 0)$

Next Steps

The next step would be to minimize the cost function, which is $350x + 200y$, within the feasible region. You can evaluate the cost at each of these vertices to find the most economical solution.

Would you like me to help find the optimal solution?

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Here are five related questions:

1. What is the cost function for the consumer's problem?
2. How do we interpret the inequality constraints in a real-world scenario?
3. How can we use the vertices of the feasible region to minimize cost?
4. What happens if the consumer needs a different proportion of AA, BB, and CC?
5. How can linear programming methods like the simplex algorithm help in this scenario?

Tip: Always verify the corner points of the feasible region before calculating the optimal solution.