document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1135433>Question 1135433</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 13. (18 pts) The cost, in dollars, for a company to produce x widgets is given by<BR>\n" );
document.write( "C(x) = 4050 + 9.00x for x ≥  0, and the price-demand function, in dollars per widget, is p(x) = 63-0.03x for 0 ≤ x ≤ 2100.\r<BR>\n" );
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document.write( "In Quiz 2, problem #10, it was seen that the profit function for this scenario isP(x) = -0.03x^2 + 54.00x-4050.\r<BR>\n" );
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document.write( "(a) The profit function is a quadratic function and so its graph is a parabola. Does the parabola open up or down? __________\r<BR>\n" );
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document.write( "(b) Find the vertex of the profit function P(x) using algebra. Show algebraic work.\r<BR>\n" );
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document.write( "(c) State the maximum profit and the number of widgets which yield that maximum profit:\r<BR>\n" );
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document.write( "The maximum profit is _____________, when ___________ widgets are produced and sold. (d) Determine the price to charge per widget in order to maximize profit.\r<BR>\n" );
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document.write( "(e) Find and interpret the break-even points. Show algebraic work. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #753060 by </B><A HREF=/tutors/aboutme.mpl?userid=MathLover1><B>MathLover1(20850)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=MathLover1><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=MathLover1><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=753060\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <BR>\n" );
document.write( "The cost, in dollars, for a company to produce x widgets is given by<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C%28x%29+=+4050+%2B+9.00x ALIGN=MIDDLE  ALT=\"C%28x%29+=+4050+%2B+9.00x\"   DISABLED_style= \"max-width:90%; height:auto;\" > for  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+%3E=++0 ALIGN=MIDDLE  ALT=\"x+%3E=++0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and \r<BR>\n" );
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document.write( "the price-demand function, in dollars per widget, is<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p%28x%29+=+63-0.03x ALIGN=MIDDLE  ALT=\"p%28x%29+=+63-0.03x\"   DISABLED_style= \"max-width:90%; height:auto;\" > for<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+0+%3C=+x+%3C=2100 ALIGN=MIDDLE  ALT=\"+0+%3C=+x+%3C=2100\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "In Quiz 2, problem #10, it was seen that the profit function for this scenario is\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28x%29+=+-0.03x%5E2+%2B+54.00x-4050 ALIGN=MIDDLE  ALT=\"P%28x%29+=+-0.03x%5E2+%2B+54.00x-4050\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "(a) The profit function is a quadratic function and so its graph is a parabola. Does the parabola open up or down? ___<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=down ALIGN=MIDDLE  ALT=\"down\"   DISABLED_style= \"max-width:90%; height:auto;\" >_______ \r<BR>\n" );
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document.write( "(b) Find the vertex of the profit function P(x) using algebra. \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28x%29+=+-0.03%28x%5E2+-54.00x%2F0.03%29-4050 ALIGN=MIDDLE  ALT=\"P%28x%29+=+-0.03%28x%5E2+-54.00x%2F0.03%29-4050\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28x%29+=+-0.03%28x%5E2+-1800x%29-4050 ALIGN=MIDDLE  ALT=\"P%28x%29+=+-0.03%28x%5E2+-1800x%29-4050\"   DISABLED_style= \"max-width:90%; height:auto;\" >...complete square<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28x%29+=+-0.03%28x%5E2+-1800x%2Bb%5E2%29-%28-%280.03%29%28b%5E2%29%29-4050 ALIGN=MIDDLE  ALT=\"P%28x%29+=+-0.03%28x%5E2+-1800x%2Bb%5E2%29-%28-%280.03%29%28b%5E2%29%29-4050\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "recall:  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2ab=1800 ALIGN=MIDDLE  ALT=\"2ab=1800\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a=1 ALIGN=MIDDLE  ALT=\"a=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, we have <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2b=1800 ALIGN=MIDDLE  ALT=\"2b=1800\"   DISABLED_style= \"max-width:90%; height:auto;\" >...solve for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b ALIGN=MIDDLE  ALT=\"b\"   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=b=900 ALIGN=MIDDLE  ALT=\"b=900\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "then we have\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28x%29+=++-+0.03+%28x%5E2+-1800x%2B900%5E2%29-%28-%280.03%29%28900%5E2%29%29-4050 ALIGN=MIDDLE  ALT=\"P%28x%29+=++-+0.03+%28x%5E2+-1800x%2B900%5E2%29-%28-%280.03%29%28900%5E2%29%29-4050\"   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=P%28x%29+=++-+0.03+%28x+-900%29%5E2%29%2B24300-4050 ALIGN=MIDDLE  ALT=\"P%28x%29+=++-+0.03+%28x+-900%29%5E2%29%2B24300-4050\"   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=P%28x%29+=++-+0.03+%28x+-+900%29%5E2%2B20250 ALIGN=MIDDLE  ALT=\"P%28x%29+=++-+0.03+%28x+-+900%29%5E2%2B20250\"   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=h=900 ALIGN=MIDDLE  ALT=\"h=900\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=20250 ALIGN=MIDDLE  ALT=\"k=20250\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "vertex is at: (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=900 ALIGN=MIDDLE  ALT=\"900\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=20250 ALIGN=MIDDLE  ALT=\"20250\"   DISABLED_style= \"max-width:90%; height:auto;\" >)\r<BR>\n" );
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document.write( "(c) State the maximum profit and the number of widgets which yield that maximum profit: <BR>\n" );
document.write( "The maximum profit is _____<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=20250 ALIGN=MIDDLE  ALT=\"20250\"   DISABLED_style= \"max-width:90%; height:auto;\" > ________, when ___<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=900 ALIGN=MIDDLE  ALT=\"x=900\"   DISABLED_style= \"max-width:90%; height:auto;\" >________ widgets are produced and sold. \r<BR>\n" );
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document.write( "(d) Determine the price to charge per widget in order to maximize profit. \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p%28x%29+=+63-0.03x ALIGN=MIDDLE  ALT=\"p%28x%29+=+63-0.03x\"   DISABLED_style= \"max-width:90%; height:auto;\" >...plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=900 ALIGN=MIDDLE  ALT=\"x=900\"   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=p%28x%29+=+63-0.03%2A900 ALIGN=MIDDLE  ALT=\"p%28x%29+=+63-0.03%2A900\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p%28x%29+=+63-27 ALIGN=MIDDLE  ALT=\"p%28x%29+=+63-27\"   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=p%28x%29+=+36 ALIGN=MIDDLE  ALT=\"p%28x%29+=+36\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "(e) Find and interpret the break-even points. Show algebraic work. \r<BR>\n" );
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document.write( "Break-Even _Point => when <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C%28x%29+=P%28x%29 ALIGN=MIDDLE  ALT=\"C%28x%29+=P%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "if<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C%28x%29+=+4050+%2B+9.00x ALIGN=MIDDLE  ALT=\"C%28x%29+=+4050+%2B+9.00x\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28x%29+=+-0.03x%5E2+%2B+54.00x-4050 ALIGN=MIDDLE  ALT=\"P%28x%29+=+-0.03x%5E2+%2B+54.00x-4050\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4050+%2B+9.00x=-0.03x%5E2+%2B+54.00x-4050 ALIGN=MIDDLE  ALT=\"4050+%2B+9.00x=-0.03x%5E2+%2B+54.00x-4050\"   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=4050+%2B+9.00x%2B0.03x%5E2+-+54.00x%2B4050=0 ALIGN=MIDDLE  ALT=\"4050+%2B+9.00x%2B0.03x%5E2+-+54.00x%2B4050=0\"   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=0.03+x%5E2+-+45+x+%2B+8100+=+0+ ALIGN=MIDDLE  ALT=\"0.03+x%5E2+-+45+x+%2B+8100+=+0+\"   DISABLED_style= \"max-width:90%; height:auto;\" >...using quadratic formula we get:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" >≈<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=209.167 ALIGN=MIDDLE  ALT=\"209.167\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" >≈<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1290.83 ALIGN=MIDDLE  ALT=\"1290.83\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );