Question 1178966: 1. Owen would like to make a small income as an artist. Owen asked his friend Emily for advice about what
combination of pictures to make. She suggested that he determine a reasonable profit for that month’s work
and then paint what he needs in order to earn that amount of profit.
• Each pastel requires $5 in materials and earns a profit of $40 for Owen.
• Each watercolor requires $15 in materials and earns a profit of $105 for Owen.
• Owen has $180 to spend on materials.
• Owen can make at most 16 pictures.
a. State the system of inequalities that represents this situation. Remember to define your variables
and include any non-negative constraints that are required. (4 marks)
b. What is the optimization equation? (1 mark)
c. On graph paper, create this feasible region to use in this problem. Label your axes. (4 marks)
d. Suppose Owen decided $1,000 would be a reasonable profit. Find three different combinations of
watercolors and pastels that would earn Owen a profit of exactly $1,000. (3 marks)
e. Now suppose Owen wanted to earn only $500 in profit. Find three different combinations of
watercolors and pastels that will earn Owen a profit of exactly $500. Using a different-coloured
pencil, add those points to your graph. (3 marks)
f. Owen’s mother has convinced him that he should try to earn as much as possible. So, Owen needs
to figure out the most profit he can earn within his constraints. He also wants to be able to prove to
his mother that it is really the maximum amount. Find the maximum possible profit that Owen can
earn and the combination of pictures he should make to earn that profit.
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Let's break down this problem step-by-step:
**a. State the System of Inequalities:**
* **Variables:**
* Let 'p' represent the number of pastels.
* Let 'w' 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.
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