document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1176641>Question 1176641</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A manufacturer of ski clothing makes ski pants and ski jackets. The profit on a pair of ski pants is $2.00 and the profit on a jacket is $1.50. Both pants and jackets require the work of sewing operators and cutters. There are 60 minutes of sewing operator time and 48 minutes of cutter time available. It takes 8 minutes to sew one pair of ski pants and 4 minutes to sew one jacket. Cutters take 4 minutes on pants and 8 minutes on a jacket.\r<BR>\n" );
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document.write( "Find the number of pants and jackets the manufacturer should make in order to maximize the profit.\r<BR>\n" );
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document.write( "  ?   pairs of pants<BR>\n" );
document.write( "  ?   jackets\r<BR>\n" );
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document.write( "What is the maximum profit? $ </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850569 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=850569\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's solve this linear programming problem step-by-step.\r<BR>\n" );
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document.write( "**1. Define Variables**\r<BR>\n" );
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document.write( "* Let 'x' be the number of ski pants.<BR>\n" );
document.write( "* Let 'y' be the number of ski jackets.\r<BR>\n" );
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document.write( "**2. Formulate the Objective Function**\r<BR>\n" );
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document.write( "The objective is to maximize profit. The profit function (P) is:\r<BR>\n" );
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document.write( "* P = 2x + 1.5y\r<BR>\n" );
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document.write( "**3. Formulate the Constraints**\r<BR>\n" );
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document.write( "* **Sewing Time Constraint:** 8x + 4y ≤ 60<BR>\n" );
document.write( "* **Cutter Time Constraint:** 4x + 8y ≤ 48<BR>\n" );
document.write( "* **Non-negativity Constraints:** x ≥ 0, y ≥ 0\r<BR>\n" );
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document.write( "**4. Simplify the Constraints**\r<BR>\n" );
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document.write( "* **Sewing Time:** 2x + y ≤ 15<BR>\n" );
document.write( "* **Cutter Time:** x + 2y ≤ 12\r<BR>\n" );
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document.write( "**5. Find the Corner Points**\r<BR>\n" );
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document.write( "* **Point 1 (Origin):** (0, 0)<BR>\n" );
document.write( "* **Point 2 (x-intercept of sewing time):** Set y = 0 in 2x + y = 15. 2x = 15, x = 7.5. (7.5, 0)<BR>\n" );
document.write( "* **Point 3 (y-intercept of cutter time):** Set x = 0 in x + 2y = 12. 2y = 12, y = 6. (0, 6)<BR>\n" );
document.write( "* **Point 4 (Intersection of the two constraints):**\r<BR>\n" );
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document.write( "   Solve the system of equations:\r<BR>\n" );
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document.write( "   * 2x + y = 15<BR>\n" );
document.write( "   * x + 2y = 12\r<BR>\n" );
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document.write( "   Multiply the second equation by 2:\r<BR>\n" );
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document.write( "   * 2x + 4y = 24\r<BR>\n" );
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document.write( "   Subtract the first equation from this:\r<BR>\n" );
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document.write( "   * 3y = 9<BR>\n" );
document.write( "   * y = 3\r<BR>\n" );
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document.write( "   Substitute y = 3 into x + 2y = 12:\r<BR>\n" );
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document.write( "   * x + 6 = 12<BR>\n" );
document.write( "   * x = 6\r<BR>\n" );
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document.write( "   Intersection point: (6, 3)\r<BR>\n" );
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document.write( "**6. Evaluate the Objective Function at Each Corner Point**\r<BR>\n" );
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document.write( "* **(0, 0):** P = 2(0) + 1.5(0) = 0<BR>\n" );
document.write( "* **(7.5, 0):** P = 2(7.5) + 1.5(0) = 15<BR>\n" );
document.write( "* **(0, 6):** P = 2(0) + 1.5(6) = 9<BR>\n" );
document.write( "* **(6, 3):** P = 2(6) + 1.5(3) = 12 + 4.5 = 16.5\r<BR>\n" );
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document.write( "**7. Determine the Maximum Profit**\r<BR>\n" );
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document.write( "The maximum profit is $16.50, which occurs at the point (6, 3).\r<BR>\n" );
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document.write( "**Answers**\r<BR>\n" );
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document.write( "* **Pairs of pants:** 6<BR>\n" );
document.write( "* **Jackets:** 3<BR>\n" );
document.write( "* **Maximum profit:** $16.50<BR>\n" );
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