document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1144750>Question 1144750</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A tent designer is working on a new tent. The tent will be made from black fabric, which costs $7 per yard, and green fabric, which costs $6 per yard. The designer will need at least 4 yards of black fabric, at least 6 yards of green fabric, and at least 12 yards of fabric overall. The total cost of the fabric used for the tent can be no more than $135. Use this information for Items 1-3. <BR>\n" );
document.write( "1. Let x represent the number of yards of black fabric and y represent the number of yards of green fabric. <BR>\n" );
document.write( "Write inequalities that model the four constraints in this situation. <BR>\n" );
document.write( "2. Graph the constraints on graph paper using a straight edge to draw the axes and lines. Be sure to label each <BR>\n" );
document.write( "axis, the units, and lines. You should have work shown to validate your shading. <BR>\n" );
document.write( "3. Identify two ordered pairs that are solutions. Explain the meaning of each ordered pairs in a complete <BR>\n" );
document.write( "sentence. <BR>\n" );
document.write( "4. What is the greatest amount of green fabric the designer can use if all the constraints are met? Explain your <BR>\n" );
document.write( "answer in a complete sentence. <BR>\n" );
document.write( "5. What is the least amount of black fabric the designer can use if all the constraints are met. Explain your <BR>\n" );
document.write( "answer in a complete sentence. <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 #766050 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=766050\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> x represents the number of yards of black fabric.<BR>\n" );
document.write( "y represents the number of yards of green fabric.\r<BR>\n" );
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document.write( "your constraints are:\r<BR>\n" );
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document.write( "x >= 4<BR>\n" );
document.write( "y >= 6<BR>\n" );
document.write( "x + y >= 12<BR>\n" );
document.write( "7x + 6y <= 135\r<BR>\n" );
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document.write( "you would graph the equality portion of these inequalities.<BR>\n" );
document.write( "specifically, you would graph:\r<BR>\n" );
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document.write( "x = 4<BR>\n" );
document.write( "y = 6<BR>\n" );
document.write( "x + y = 12<BR>\n" );
document.write( "7x + 6y = 135\r<BR>\n" );
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document.write( "you would then shade the area of the graph that satisfies the inequalities.\r<BR>\n" );
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document.write( "your graph would look like this.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2019/091001.jpg\" alt=\"$$$\" >\r<BR>\n" );
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document.write( "your region of feasibility would be the area......\r<BR>\n" );
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document.write( "to the right of x = 4<BR>\n" );
document.write( "above y = 6<BR>\n" );
document.write( "above x + y = 12<BR>\n" );
document.write( "below 7x + 6y = 135\r<BR>\n" );
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document.write( "this corresponds to your constraint inequalities shown above.\r<BR>\n" );
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document.write( "the corner points of the feasible region are the points where your maximum or minimum values for the objective function would be.\r<BR>\n" );
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document.write( "these corner points are where the solution to your problem lies.\r<BR>\n" );
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document.write( "this problem does not have an objective function, so you would not do an evaluation at any of the corner points.\r<BR>\n" );
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document.write( "one objective function might be to minimize your costs.<BR>\n" );
document.write( "in that case, your objective function would be cost = 7x + 6y.<BR>\n" );
document.write( "you would evaluate your objective function at each of the corner points and then pick the corner point that has the least cost.\r<BR>\n" );
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document.write( "the answer to the questions that you do have are shown below.\r<BR>\n" );
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document.write( "1. Let x represent the number of yards of black fabric and y represent the number of yards of green fabric.<BR>\n" );
document.write( "Write inequalities that model the four constraints in this situation.\r<BR>\n" );
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document.write( "done above.\r<BR>\n" );
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document.write( "2. Graph the constraints on graph paper using a straight edge to draw the axes and lines. Be sure to label each<BR>\n" );
document.write( "axis, the units, and lines. You should have work shown to validate your shading.\r<BR>\n" );
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document.write( "done above.\r<BR>\n" );
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document.write( "3. Identify two ordered pairs that are solutions. Explain the meaning of each ordered pairs in a complete<BR>\n" );
document.write( "sentence.\r<BR>\n" );
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document.write( "(4,8) is one solution.<BR>\n" );
document.write( "it satisfies the constraints as follows:<BR>\n" );
document.write( "x >= 4<BR>\n" );
document.write( "y >= 6<BR>\n" );
document.write( "x + y = 4 + 8 = 12 >= 12<BR>\n" );
document.write( "7x + 6y = 7 * 4 + 6 * 8 = 28 + 48 = 76 <= 135\r<BR>\n" );
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document.write( "(14.143,6) is another solution.<BR>\n" );
document.write( "it satisfies the constraints as follows:<BR>\n" );
document.write( "x >= 4<BR>\n" );
document.write( "y >= 6<BR>\n" );
document.write( "x + y = 14.143 + 6 = 20.143 >= 12<BR>\n" );
document.write( "7x + 6y = 7 * 14.143 + 6 * 6 = 135 <= 135 (true if you take away the rounding to 3 decimal places that the graph shows - un-rounded value would be approximately 14.142857 as found on my TI-84 Plus).\r<BR>\n" );
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document.write( "4. What is the greatest amount of green fabric the designer can use if all the constraints are met? Explain your answer in a complete sentence.\r<BR>\n" );
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document.write( "the greatest amount of green fabric would be at the point (4,17.833).<BR>\n" );
document.write( "that where the value of y is the greatest.<BR>\n" );
document.write( "at that point, the constraint that limits the value of y to be greater is the constraint of 7x + 6y <= 135.\r<BR>\n" );
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document.write( "5. What is the least amount of black fabric the designer can use if all the constraints are met. Explain your answer in a complete sentence.\r<BR>\n" );
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document.write( "the least amount of black fabric would be at the points (4,8) and (4,17.833).<BR>\n" );
document.write( "that's where the value of x is the smallest.<BR>\n" );
document.write( "that satisfies the constraint that x >= 4.<BR>\n" );
document.write( "(4,17.833) also satisfies the constraint that 7x + 6y <= 135.<BR>\n" );
document.write( "(4,8) also satisfies the constraint that x + y >= 12.\r<BR>\n" );
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document.write( "you basically satisfies all the constraint inequalities at each of the corner points.<BR>\n" );
document.write( "all of the corner points should satisfy all of the constraints.<BR>\n" );
document.write( "in this graph, they do.<BR>\n" );
document.write( "if any of the constraints are not satisfied at a particular corner point, then that point is invalid.\r<BR>\n" );
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