<PRE><B>Question 1178966</B>
Let&#39;s break down this problem step-by-step:

**a. State the System of Inequalities:**

* **Variables:**
    * Let &#39;p&#39; represent the number of pastels.
    * Let &#39;w&#39; represent the number of watercolors.
* **Material Cost Constraint:**
    * 5p + 15w ≤ 180 (Owen has $180 to spend on materials)
* **Total Pictures Constraint:**
    * p + w ≤ 16 (Owen can make at most 16 pictures)
* **Non-Negative Constraints:**
    * p ≥ 0 (Owen cannot make a negative number of pastels)
    * w ≥ 0 (Owen cannot make a negative number of watercolors)

**b. What is the Optimization Equation?**

* **Profit Equation:**
    * Profit = 40p + 105w (Owen earns $40 per pastel and $105 per watercolor)

**c. Create the Feasible Region (Graph):**

1.  **Graph 5p + 15w ≤ 180:**
    * Rewrite as: p + 3w ≤ 36
    * Find intercepts:
        * If p = 0, 3w = 36, w = 12 (point: (0, 12))
        * If w = 0, p = 36 (point: (36, 0))
    * Draw a line through these points. Shade below the line.
2.  **Graph p + w ≤ 16:**
    * Find intercepts:
        * If p = 0, w = 16 (point: (0, 16))
        * If w = 0, p = 16 (point: (16, 0))
    * Draw a line through these points. Shade below the line.
3.  **Graph p ≥ 0 and w ≥ 0:**
    * This restricts the feasible region to the first quadrant.
4.  **Feasible Region:** The area where all shaded regions overlap.
    * Find the corner points where the lines intersect.
        * (0, 0)
        * (16, 0)
        * (0, 12)
        * Intersection of p + 3w = 36 and p + w = 16:
            * Subtract the equations: 2w = 20, w = 10
            * Substitute w = 10 into p + w = 16: p = 6 (point: (6, 10))

**d. Combinations for $1,000 Profit:**

* 1000 = 40p + 105w
* We need to find integer solutions for p and w.
* 1000/5 = 200, so 40p + 105w must be divisible by 5. 105w is always divisible by 5, so 40p must be divisible by 5.

    * **Combination 1:** If w = 0, 40p = 1000, p = 25. (25, 0)
    * **Combination 2:** If w = 4, 40p + 420 = 1000, 40p = 580, p = 14.5. (not integer)
    * **Combination 3:** If w = 8, 40p + 840 = 1000, 40p = 160, p = 4. (4, 8)
    * **Combination 4:** if w=12, 105w=1260. too high.
    * **Combination 5:** if p = 10, 400 + 105w = 1000, 105w = 600. w= 5.7. not integer.
    * **Combination 6:** if p=20, 800+105w=1000, 105w=200, not integer.
    * Therefore: (25, 0), (4, 8), (10, 5.7) is not a solution.

    * (25, 0)
    * (4, 8)
    * (10, 5.7) is not a solution, but (10, 6) will be slightly over 1000.

**e. Combinations for $500 Profit:**

* 500 = 40p + 105w
* We need to find integer solutions for p and w.

    * **Combination 1:** If w = 0, 40p = 500, p = 12.5. (not integer)
    * **Combination 2:** If w = 2, 40p + 210 = 500, 40p = 290, p = 7.25. (not integer)
    * **Combination 3:** If w = 4, 40p + 420 = 500, 40p = 80, p = 2. (2, 4)
    * **Combination 4:** If p = 5, 200 + 105w = 500, 105w = 300, w = 2.85. not integer.
    * **Combination 5:** If p = 10, 400 + 105w = 500, 105w = 100, not integer.
    * Therefore: (2, 4), (12.5, 0), (5, 2.85) is not a solution.

    * (2, 4)
    * (12.5,0)
    * (5, 2.85) is not a solution.

**f. Maximum Profit:**

* Evaluate the profit equation at the corner points of the feasible region:
    * (0, 0): Profit = 40(0) + 105(0) = 0
    * (16, 0): Profit = 40(16) + 105(0) = 640
    * (0, 12): Profit = 40(0) + 105(12) = 1260
    * (6, 10): Profit = 40(6) + 105(10) = 240 + 1050 = 1290

* **Maximum Profit:** $1290
* **Combination:** 6 pastels and 10 watercolors.
</PRE>