document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1189926>Question 1189926</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Given:<BR>\n" );
document.write( "Room1 : length=4m : width=3m<BR>\n" );
document.write( "Room2 : length=6m : width=4m<BR>\n" );
document.write( "Tiles:<BR>\n" );
document.write( "Glossy1 : 0.5m x 0.5m cost $8/tile<BR>\n" );
document.write( "Glossy2 : 1m x 1m     cost $10/tile<BR>\n" );
document.write( "Non glossy1: 0.5m x 0.5m cost $5/tile<BR>\n" );
document.write( "Non glossy2: 1m x 1m     cost $7/tile<BR>\n" );
document.write( "How to formulate 2 quadratic inequalities involving the dimensions of the floor of the rooms and the measure and costs of the tiles. Then graph the solution sets of these inequalities.\r<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849259 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=849259\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to formulate the quadratic inequalities and approach the graphing:\r<BR>\n" );
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document.write( "**1. Area Calculations:**\r<BR>\n" );
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document.write( "*   **Room 1 Area:** Length * Width = 4m * 3m = 12 m²<BR>\n" );
document.write( "*   **Room 2 Area:** Length * Width = 6m * 4m = 24 m²\r<BR>\n" );
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document.write( "**2. Tile Area Calculations:**\r<BR>\n" );
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document.write( "*   **Glossy 1 Tile Area:** 0.5m * 0.5m = 0.25 m²<BR>\n" );
document.write( "*   **Glossy 2 Tile Area:** 1m * 1m = 1 m²<BR>\n" );
document.write( "*   **Non-Glossy 1 Tile Area:** 0.5m * 0.5m = 0.25 m²<BR>\n" );
document.write( "*   **Non-Glossy 2 Tile Area:** 1m * 1m = 1 m²\r<BR>\n" );
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document.write( "**3. Number of Tiles Needed (Variables):**\r<BR>\n" );
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document.write( "Let's define variables for the number of each type of tile:\r<BR>\n" );
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document.write( "*   g1 = Number of Glossy 1 tiles<BR>\n" );
document.write( "*   g2 = Number of Glossy 2 tiles<BR>\n" );
document.write( "*   n1 = Number of Non-Glossy 1 tiles<BR>\n" );
document.write( "*   n2 = Number of Non-Glossy 2 tiles\r<BR>\n" );
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document.write( "**4. Area Inequalities:**\r<BR>\n" );
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document.write( "Since the total tile area must be *at least* equal to the room area, we have two inequalities for each room:\r<BR>\n" );
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document.write( "*   **Room 1:**<BR>\n" );
document.write( "    *   0.25g1 + 1g2 + 0.25n1 + 1n2 ≥ 12<BR>\n" );
document.write( "*   **Room 2:**<BR>\n" );
document.write( "    *   0.25g1 + 1g2 + 0.25n1 + 1n2 ≥ 24\r<BR>\n" );
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document.write( "**5. Cost Inequalities:**\r<BR>\n" );
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document.write( "Let's assume a combined budget for both rooms. Let 'B' be the total budget. The cost inequalities are:\r<BR>\n" );
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document.write( "*   8g1 + 10g2 + 5n1 + 7n2 ≤ B\r<BR>\n" );
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document.write( "**6. Graphing the Solution Sets:**\r<BR>\n" );
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document.write( "Graphing these inequalities directly in 4D space (g1, g2, n1, n2) is impossible on a 2D surface.  To visualize, we need to make some simplifications.  Here are a few approaches:\r<BR>\n" );
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document.write( "*   **Scenario 1: Fixed Tile Mix:** Assume a fixed ratio of tile types (e.g., equal numbers of glossy and non-glossy tiles). This reduces the variables and makes graphing possible.  For example, if g1 = n1 and g2 = n2, our inequalities would be:\r<BR>\n" );
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document.write( "    *   Room 1: 0.5g1 + 2g2 ≥ 12<BR>\n" );
document.write( "    *   Room 2: 0.5g1 + 2g2 ≥ 24<BR>\n" );
document.write( "    *   Cost: 13g1 + 17g2 ≤ B\r<BR>\n" );
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document.write( "    Now you have inequalities in 2D space (g1 and g2) that you can graph.\r<BR>\n" );
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document.write( "*   **Scenario 2: Focus on One Room, Two Tile Types:** Consider just Room 1 and two tile types (e.g., Glossy 1 and Glossy 2).  The inequalities become:\r<BR>\n" );
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document.write( "    *   0.25g1 + g2 ≥ 12<BR>\n" );
document.write( "    *   8g1 + 10g2 ≤ B (budget for Room 1)\r<BR>\n" );
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document.write( "    Again, this is a 2D graphing problem (g1 and g2).\r<BR>\n" );
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document.write( "*   **Software:** Software like GeoGebra or Wolfram Alpha can handle 3D graphing, which could be used if you can reduce the number of variables to three.\r<BR>\n" );
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document.write( "**7. Interpreting the Graphs:**\r<BR>\n" );
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document.write( "The solution set (the feasible region) on the graph represents the combinations of tile numbers that satisfy both the area and cost constraints.  Any point within this region represents a possible solution.\r<BR>\n" );
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document.write( "**Example (Scenario 2):**\r<BR>\n" );
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document.write( "Let's say the budget for Room 1 is $200. Our inequalities are:\r<BR>\n" );
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document.write( "*   0.25g1 + g2 ≥ 12<BR>\n" );
document.write( "*   8g1 + 10g2 ≤ 200\r<BR>\n" );
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document.write( "You would graph these two inequalities on a g1-g2 plane. The overlapping shaded region is your solution set.\r<BR>\n" );
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document.write( "**Important Considerations:**\r<BR>\n" );
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document.write( "*   **Integer Solutions:**  Since you can't buy fractions of tiles, you're only interested in integer solutions within the feasible region.<BR>\n" );
document.write( "*   **Optimization:**  You could add an objective function (e.g., minimize cost) and use linear programming techniques to find the optimal solution.<BR>\n" );
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