document.write( "Question 56274: A home-based company produces both hand-knitted scarves and sweaters. The scarves take 2 hours of labor to produce, and the sweaters take 14 hours. The labor available is limited to 40 hours per week, and the total production capacity is 5 items per week. Write a system of inequalities representing this situation, where x is the number of scarves and y is the number of sweaters. Then graph the system of inequalities. THANK YOU IN ADVANCE \n" ); document.write( "

Algebra.Com's Answer #38175 by Edwin McCravy(20056) $\"\"$ $\"About$

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A home-based company produces both hand-knitted scarves and sweaters.\r\n" );
document.write( "The scarves take 2 hours of labor to produce, and the sweaters take 14 hours.\r\n" );
document.write( "The labor available is limited to 40 hours per week, and the total production\r\n" );
document.write( "capacity is 5 items per week.  Write a system of inequalities representing\r\n" );
document.write( "this situation, where x is the number of scarves and y is the number of\r\n" );
document.write( "sweaters.  Then graph the system of inequalities. THANK YOU IN ADVANCE\r\n" );
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document.write( "Each scarf takes 2 hours of labor. Therefore x scarves takes 2x hours of\r\n" );
document.write( "labor. Each sweater takes 14 hours of labor. Therefore x sweaters takes 14y\r\n" );
document.write( "hours of labor\r\n" );
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document.write( "Therefore the total number of hours of labor is 2x + 14y.\r\n" );
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document.write( "Since the number of hours of labor must be less than or equal to 40, we \r\n" );
document.write( "have the inequality:\r\n" );
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document.write( "         2x + 14y < 40\r\n" );
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document.write( "Since they are limited to x items per week,\r\n" );
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document.write( "            x + y < 5\r\n" );
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document.write( "Now there are two additional inequalities, the obvious ones, which\r\n" );
document.write( "indicate that the number of each item cannot be negative.  These are\r\n" );
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document.write( "                x > 0\r\n" );
document.write( "                y > 0\r\n" );
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document.write( "These last two limit the graph to the upper right hand region of\r\n" );
document.write( "the xy-plane.  So we draw the graphs of the lines we get by \r\n" );
document.write( "replacing the inequalities by equal signs.\r\n" );
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document.write( "2x + 14y = 40\r\n" );
document.write( "   x + y = 5\r\n" );
document.write( "       x = 0  (the y axis)\r\n" );
document.write( "       y = 0  (the x axis)\r\n" );
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document.write( "Shade the region which is below both the red and the green \r\n" );
document.write( "lines, which is above the x-axis and to the right of the \r\n" );
document.write( "y-axis. This is the feasible region.\r\n" );
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document.write( "Edwin McCravy

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