document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1192771>Question 1192771</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A widget manufacturer estimates that her monthly revenue can be modeled by the function R(x)=-0.006x^2+32x-10,000. What is the minimum number of items that must be sold for the revenue to equal $30,000? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #824664 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=824664\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Method 1:<BR>\n" );
document.write( "Use your favorite graphing tool to plot the two equations<BR>\n" );
document.write( "y = -0.006x^2+32x-10,000<BR>\n" );
document.write( "y = 30,000\r<BR>\n" );
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document.write( "Here's what it looks like in Desmos which is a free online graphing calculator<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://www.desmos.com/calculator/tnzwh3datm\">https://www.desmos.com/calculator/tnzwh3datm</a><BR>\n" );
document.write( "Take note of how the x and y axis scale is set up.\r<BR>\n" );
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document.write( "After plotting both the parabolic curve and straight line, click on where they intersect or cross.<BR>\n" );
document.write( "Desmos will display the (x,y) coordinates of each intersection point.\r<BR>\n" );
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document.write( "The two intersection points are:<BR>\n" );
document.write( "(2000, 30000)<BR>\n" );
document.write( "(3333.333, 30000) approximately<BR>\n" );
document.write( "When doing ordered pair notation, do NOT have a comma in the number \"30000\" aka \"thirty thousand\". This is because the comma is used to separate the x and y coordinates.\r<BR>\n" );
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document.write( "The x coordinate of the left-most point is what we're after.<BR>\n" );
document.write( "This is the smaller x value such that y = 30,000.<BR>\n" );
document.write( "In other words, it's the minimum amount of items to sell to achieve a revenue of $30,000.\r<BR>\n" );
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document.write( "Answer: <font color=red>2000 units</font>.\r<BR>\n" );
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document.write( "Method 2:<BR>\n" );
document.write( "Instead of graphing two equations, we can graph one equation.\r<BR>\n" );
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document.write( "Since we want the revenue R(x) to be $30,000, this means:<BR>\n" );
document.write( "R(x) = 30,000<BR>\n" );
document.write( "-0.006x^2+32x-10,000 = 30,000<BR>\n" );
document.write( "-0.006x^2+32x-10,000 - 30,000 = 0<BR>\n" );
document.write( "-0.006x^2+32x-40,000 = 0\r<BR>\n" );
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document.write( "The single equation to graph is<BR>\n" );
document.write( "y = -0.006x^2+32x-40,000\r<BR>\n" );
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document.write( "When graphing this, we're looking for the x intercepts which make y = 0. Therefore, we're looking where the curve either crosses the x axis or touches it to bounce away.\r<BR>\n" );
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document.write( "Graph:<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://www.desmos.com/calculator/vuz7masgvd\">https://www.desmos.com/calculator/vuz7masgvd</a><BR>\n" );
document.write( "As the graph shows, the x coordinates of the x intercepts are the same as the results of method 1. The only difference is now that y = 0 instead of y = 30,000.\r<BR>\n" );
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document.write( "We'll arrive at the same final answer of <font color=red>2000 units</font> after selecting the smaller root.\r<BR>\n" );
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document.write( "Method 3:<BR>\n" );
document.write( "This method won't use any graphs. Instead, I'll use the quadratic formula.\r<BR>\n" );
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document.write( "R(x) = 30,000<BR>\n" );
document.write( "-0.006x^2+32x-10,000 = 30,000<BR>\n" );
document.write( "-0.006x^2+32x-10,000 - 30,000 = 0<BR>\n" );
document.write( "-0.006x^2+32x-40,000 = 0\r<BR>\n" );
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document.write( "That final equation is in the form ax^2+bx+c = 0 where,<BR>\n" );
document.write( "a = -0.006<BR>\n" );
document.write( "b = 32<BR>\n" );
document.write( "c = -40,000\r<BR>\n" );
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document.write( "Plugging those three items into the quadratic formula yields:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-%2832%29%2B-sqrt%28%2832%29%5E2-4%28-0.006%29%28-40000%29%29%29%2F%282%28-0.006%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-32%2B-sqrt%2864%29%29%2F%28-0.012%29 ALIGN=MIDDLE  ALT=\"x+=+%28-32%2B-sqrt%2864%29%29%2F%28-0.012%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-32%2B-+++8%29%2F%28-0.012%29 ALIGN=MIDDLE  ALT=\"x+=+%28-32%2B-+++8%29%2F%28-0.012%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-32%2B8%29%2F%28-0.012%29 ALIGN=MIDDLE  ALT=\"x+=+%28-32%2B8%29%2F%28-0.012%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-32-8%29%2F%28-0.012%29 ALIGN=MIDDLE  ALT=\"x+=+%28-32-8%29%2F%28-0.012%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-24%29%2F%28-0.012%29 ALIGN=MIDDLE  ALT=\"x+=+%28-24%29%2F%28-0.012%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-40%29%2F%28-0.012%29 ALIGN=MIDDLE  ALT=\"x+=+%28-40%29%2F%28-0.012%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+2000 ALIGN=MIDDLE  ALT=\"x+=+2000\"   DISABLED_style= \"max-width:90%; height:auto;\" > or  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+3333.333 ALIGN=MIDDLE  ALT=\"x+=+3333.333\"   DISABLED_style= \"max-width:90%; height:auto;\" > approximately\r<BR>\n" );
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document.write( "Like before, we pick the smaller x solution to get <font color=red>2000 units</font> as the final answer.<BR>\n" );
document.write( "</font> </SPAN>\n" );
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