document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1062038>Question 1062038</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Good evening, can you please help me solve the 3rd one? Thank you very much!\r<BR>\n" );
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document.write( "(1) An office manager needs to staff the office. She hires full-time employees at $18 per our and part-time employees at $12 per hour. Write an objective function that represents the total cost (in $) to staff the office with x full-time employees and y part-time employees for 1 hr.<BR>\n" );
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document.write( "ANSWER: Total cost: z = 18x + 12y<BR>\n" );
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document.write( "(2) Refer to 1. Suppose that the office manager needs at least 20 employees, but not more than 24 full-time employees. Furthermore, to make the office run smoothly, the manager knows that the number of full-time employees must always be greater than or equal to the number of part-time employees. Write a system of inequalities that represents the constraints on the number of full-time employees x and the number of part-time employees y.<BR>\n" );
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document.write( "ANSWER:  x+y&#8805;20<BR>\n" );
document.write( "                x&#8804;24<BR>\n" );
document.write( "                x&#8805;y<BR>\n" );
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document.write( "(3) Refer to no's 1 and 2. The office manager needs at least 20 employees, but not more than 24 full-time employees. Furthermore, to make the office run smoothly, the manager knows that the number of full-time employees must always be greater than or equal to the number of part-time employees. If she pays full-time employees $18 per hour and part-time employees $12 per hour, determine the number of full-time and part-time employees she should hire to minimize total labor cost per hour. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #676800 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=676800\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> you graph the constraints and then you evaluate the objective function at the corners of the feasible region.\r<BR>\n" );
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document.write( "in the attached graph, the feasible region is the region that is NOT shaded.\r<BR>\n" );
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document.write( "here's the graph:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2016/121901.jpg\" alt=\"%%%\" </>\r<BR>\n" );
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document.write( "the corner points are:\r<BR>\n" );
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document.write( "(10,10)<BR>\n" );
document.write( "(20,0)<BR>\n" );
document.write( "(24,0)<BR>\n" );
document.write( "(24,24)\r<BR>\n" );
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document.write( "evaluate the objective function at these corner points and the minimum cost is when 10 full time employers are hired and 10 part time employees are hired.\r<BR>\n" );
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document.write( "the objective function is the cost function of z = 18x + 12y\r<BR>\n" );
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document.write( "at the coordinate point of (10,10), the cost is 180 + 120 = 400.\r<BR>\n" );
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document.write( "the constraints at (10,10) are satisfied.\r<BR>\n" );
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document.write( "x + y >= 20<BR>\n" );
document.write( "x <= 24<BR>\n" );
document.write( "x >= y\r<BR>\n" );
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document.write( "the total constraints are:\r<BR>\n" );
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document.write( "x + y >= 20<BR>\n" );
document.write( "x <= 24<BR>\n" );
document.write( "x >= y<BR>\n" );
document.write( "x >= 0<BR>\n" );
document.write( "y >= 0\r<BR>\n" );
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document.write( "the last two are necessary because neither x nor y can be less than 0.\r<BR>\n" );
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document.write( "in the graph, i shaded the regions that do NOT satisfy the constraints.<BR>\n" );
document.write( "what is left is the region that DOES satisfy the constraints.\r<BR>\n" );
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document.write( "you still need to pay attention to the original constraints since that tells you what your answer can be.\r<BR>\n" );
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document.write( "for example, if the constraint was x + y > 20, then (10,10) would not satisfy that constraint because it is not greater than 20.\r<BR>\n" );
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document.write( "(10,10) would still be a marker that tells you the area where the answer might lie.\r<BR>\n" );
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document.write( "(11,10) is still in the region of feasibility, as is (10,11).  <BR>\n" );
document.write( "either one of those would suffice if the requirement was > rather than >=.\r<BR>\n" );
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document.write( "in fact (10,11) would be the least cost solution in that case, since the pay of the temporary employee is less than the pay of the regular employee.\r<BR>\n" );
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document.write( "the software to graph that i used is at www.desmos.com\r<BR>\n" );
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document.write( "here's a reference on graphing linear inequalities.\r<BR>\n" );
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document.write( "<a  rel=nofollow HREF= \"http://www.purplemath.com/modules/ineqgrph.htm\" target= \"_blank\">http://www.purplemath.com/modules/ineqgrph.htm</a>\r<BR>\n" );
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document.write( "note that they tell you to shade the region of feasibility and that to use dashed lines if the inequality does not contain an equal as part of it.\r<BR>\n" );
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document.write( "i did the opposite only because i was not manually creating the graph and found that not shading the region of feasibility allowed it to show up better when using desmos software to generate the graph.\r<BR>\n" );
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document.write( "the general idea is that you graph the region of feasibility and then look for the corner points of that region.\r<BR>\n" );
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document.write( "your minimum / maximum solution, if there is one, should be at those corner points.\r<BR>\n" );
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document.write( "here's another reference that addresses corner points.\r<BR>\n" );
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document.write( "<a  rel=nofollow HREF= \"http://www.purplemath.com/modules/linprog.htm\" target = \"_blank\">http://www.purplemath.com/modules/linprog.htm</a> </SPAN>\n" );
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