document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1178966>Question 1178966</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 1. Owen would like to make a small income as an artist. Owen asked his friend Emily for advice about what<BR>\n" );
document.write( "combination of pictures to make. She suggested that he determine a reasonable profit for that month’s work<BR>\n" );
document.write( "and then paint what he needs in order to earn that amount of profit.<BR>\n" );
document.write( "• Each pastel requires $5 in materials and earns a profit of $40 for Owen.<BR>\n" );
document.write( "• Each watercolor requires $15 in materials and earns a profit of $105 for Owen.<BR>\n" );
document.write( "• Owen has $180 to spend on materials.<BR>\n" );
document.write( "• Owen can make at most 16 pictures.<BR>\n" );
document.write( "a. State the system of inequalities that represents this situation. Remember to define your variables<BR>\n" );
document.write( "and include any non-negative constraints that are required. (4 marks)<BR>\n" );
document.write( "b. What is the optimization equation? (1 mark)<BR>\n" );
document.write( "c. On graph paper, create this feasible region to use in this problem. Label your axes. (4 marks)<BR>\n" );
document.write( "d. Suppose Owen decided $1,000 would be a reasonable profit. Find three different combinations of<BR>\n" );
document.write( "watercolors and pastels that would earn Owen a profit of exactly $1,000. (3 marks)<BR>\n" );
document.write( "e. Now suppose Owen wanted to earn only $500 in profit. Find three different combinations of<BR>\n" );
document.write( "watercolors and pastels that will earn Owen a profit of exactly $500. Using a different-coloured<BR>\n" );
document.write( "pencil, add those points to your graph. (3 marks)<BR>\n" );
document.write( "f. Owen’s mother has convinced him that he should try to earn as much as possible. So, Owen needs<BR>\n" );
document.write( "to figure out the most profit he can earn within his constraints. He also wants to be able to prove to<BR>\n" );
document.write( "his mother that it is really the maximum amount. Find the maximum possible profit that Owen can<BR>\n" );
document.write( "earn and the combination of pictures he should make to earn that profit.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850272 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=850272\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's break down this problem step-by-step:\r<BR>\n" );
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document.write( "**a. State the System of Inequalities:**\r<BR>\n" );
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document.write( "* **Variables:**<BR>\n" );
document.write( "    * Let 'p' represent the number of pastels.<BR>\n" );
document.write( "    * Let 'w' represent the number of watercolors.<BR>\n" );
document.write( "* **Material Cost Constraint:**<BR>\n" );
document.write( "    * 5p + 15w ≤ 180 (Owen has $180 to spend on materials)<BR>\n" );
document.write( "* **Total Pictures Constraint:**<BR>\n" );
document.write( "    * p + w ≤ 16 (Owen can make at most 16 pictures)<BR>\n" );
document.write( "* **Non-Negative Constraints:**<BR>\n" );
document.write( "    * p ≥ 0 (Owen cannot make a negative number of pastels)<BR>\n" );
document.write( "    * w ≥ 0 (Owen cannot make a negative number of watercolors)\r<BR>\n" );
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document.write( "**b. What is the Optimization Equation?**\r<BR>\n" );
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document.write( "* **Profit Equation:**<BR>\n" );
document.write( "    * Profit = 40p + 105w (Owen earns $40 per pastel and $105 per watercolor)\r<BR>\n" );
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document.write( "**c. Create the Feasible Region (Graph):**\r<BR>\n" );
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document.write( "1.  **Graph 5p + 15w ≤ 180:**<BR>\n" );
document.write( "    * Rewrite as: p + 3w ≤ 36<BR>\n" );
document.write( "    * Find intercepts:<BR>\n" );
document.write( "        * If p = 0, 3w = 36, w = 12 (point: (0, 12))<BR>\n" );
document.write( "        * If w = 0, p = 36 (point: (36, 0))<BR>\n" );
document.write( "    * Draw a line through these points. Shade below the line.<BR>\n" );
document.write( "2.  **Graph p + w ≤ 16:**<BR>\n" );
document.write( "    * Find intercepts:<BR>\n" );
document.write( "        * If p = 0, w = 16 (point: (0, 16))<BR>\n" );
document.write( "        * If w = 0, p = 16 (point: (16, 0))<BR>\n" );
document.write( "    * Draw a line through these points. Shade below the line.<BR>\n" );
document.write( "3.  **Graph p ≥ 0 and w ≥ 0:**<BR>\n" );
document.write( "    * This restricts the feasible region to the first quadrant.<BR>\n" );
document.write( "4.  **Feasible Region:** The area where all shaded regions overlap.<BR>\n" );
document.write( "    * Find the corner points where the lines intersect.<BR>\n" );
document.write( "        * (0, 0)<BR>\n" );
document.write( "        * (16, 0)<BR>\n" );
document.write( "        * (0, 12)<BR>\n" );
document.write( "        * Intersection of p + 3w = 36 and p + w = 16:<BR>\n" );
document.write( "            * Subtract the equations: 2w = 20, w = 10<BR>\n" );
document.write( "            * Substitute w = 10 into p + w = 16: p = 6 (point: (6, 10))\r<BR>\n" );
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document.write( "**d. Combinations for $1,000 Profit:**\r<BR>\n" );
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document.write( "* 1000 = 40p + 105w<BR>\n" );
document.write( "* We need to find integer solutions for p and w.<BR>\n" );
document.write( "* 1000/5 = 200, so 40p + 105w must be divisible by 5. 105w is always divisible by 5, so 40p must be divisible by 5.\r<BR>\n" );
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document.write( "    * **Combination 1:** If w = 0, 40p = 1000, p = 25. (25, 0)<BR>\n" );
document.write( "    * **Combination 2:** If w = 4, 40p + 420 = 1000, 40p = 580, p = 14.5. (not integer)<BR>\n" );
document.write( "    * **Combination 3:** If w = 8, 40p + 840 = 1000, 40p = 160, p = 4. (4, 8)<BR>\n" );
document.write( "    * **Combination 4:** if w=12, 105w=1260. too high.<BR>\n" );
document.write( "    * **Combination 5:** if p = 10, 400 + 105w = 1000, 105w = 600. w= 5.7. not integer.<BR>\n" );
document.write( "    * **Combination 6:** if p=20, 800+105w=1000, 105w=200, not integer.<BR>\n" );
document.write( "    * Therefore: (25, 0), (4, 8), (10, 5.7) is not a solution.\r<BR>\n" );
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document.write( "    * (25, 0)<BR>\n" );
document.write( "    * (4, 8)<BR>\n" );
document.write( "    * (10, 5.7) is not a solution, but (10, 6) will be slightly over 1000.\r<BR>\n" );
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document.write( "**e. Combinations for $500 Profit:**\r<BR>\n" );
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document.write( "* 500 = 40p + 105w<BR>\n" );
document.write( "* We need to find integer solutions for p and w.\r<BR>\n" );
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document.write( "    * **Combination 1:** If w = 0, 40p = 500, p = 12.5. (not integer)<BR>\n" );
document.write( "    * **Combination 2:** If w = 2, 40p + 210 = 500, 40p = 290, p = 7.25. (not integer)<BR>\n" );
document.write( "    * **Combination 3:** If w = 4, 40p + 420 = 500, 40p = 80, p = 2. (2, 4)<BR>\n" );
document.write( "    * **Combination 4:** If p = 5, 200 + 105w = 500, 105w = 300, w = 2.85. not integer.<BR>\n" );
document.write( "    * **Combination 5:** If p = 10, 400 + 105w = 500, 105w = 100, not integer.<BR>\n" );
document.write( "    * Therefore: (2, 4), (12.5, 0), (5, 2.85) is not a solution.\r<BR>\n" );
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document.write( "    * (2, 4)<BR>\n" );
document.write( "    * (12.5,0)<BR>\n" );
document.write( "    * (5, 2.85) is not a solution.\r<BR>\n" );
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document.write( "**f. Maximum Profit:**\r<BR>\n" );
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document.write( "* Evaluate the profit equation at the corner points of the feasible region:<BR>\n" );
document.write( "    * (0, 0): Profit = 40(0) + 105(0) = 0<BR>\n" );
document.write( "    * (16, 0): Profit = 40(16) + 105(0) = 640<BR>\n" );
document.write( "    * (0, 12): Profit = 40(0) + 105(12) = 1260<BR>\n" );
document.write( "    * (6, 10): Profit = 40(6) + 105(10) = 240 + 1050 = 1290\r<BR>\n" );
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document.write( "* **Maximum Profit:** $1290<BR>\n" );
document.write( "* **Combination:** 6 pastels and 10 watercolors.<BR>\n" );
document.write( " </SPAN>\n" );
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