document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1171433>Question 1171433</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Elmer is organizing a fund raising basketball tournament in their barangay. He plans to charge PHP20 entry fee for each 80 players. He recently decided to raise the entry fee by PHP5 and 5 fewer players entered with the increase. How much would Elmer charge in order to maximize the income? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850934 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><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=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=850934\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's solve this optimization problem step-by-step.\r<BR>\n" );
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document.write( "**1. Define Variables**\r<BR>\n" );
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document.write( "* Let 'x' be the number of PHP5 increases in the entry fee.<BR>\n" );
document.write( "* Entry fee = 20 + 5x<BR>\n" );
document.write( "* Number of players = 80 - 5x<BR>\n" );
document.write( "* Income (I) = (Entry fee) * (Number of players)\r<BR>\n" );
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document.write( "**2. Formulate the Income Function**\r<BR>\n" );
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document.write( "* I(x) = (20 + 5x)(80 - 5x)<BR>\n" );
document.write( "* I(x) = 1600 - 100x + 400x - 25x^2<BR>\n" );
document.write( "* I(x) = -25x^2 + 300x + 1600\r<BR>\n" );
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document.write( "**3. Find the Vertex of the Quadratic Function**\r<BR>\n" );
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document.write( "The income function is a quadratic function, and its graph is a parabola that opens downward. The maximum income occurs at the vertex of the parabola.\r<BR>\n" );
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document.write( "The x-coordinate of the vertex is given by:\r<BR>\n" );
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document.write( "* x = -b / (2a)\r<BR>\n" );
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document.write( "Where:\r<BR>\n" );
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document.write( "* a = -25<BR>\n" );
document.write( "* b = 300\r<BR>\n" );
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document.write( "Plugging in the values:\r<BR>\n" );
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document.write( "* x = -300 / (2 * -25)<BR>\n" );
document.write( "* x = -300 / -50<BR>\n" );
document.write( "* x = 6\r<BR>\n" );
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document.write( "**4. Calculate the Entry Fee**\r<BR>\n" );
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document.write( "Now, substitute the value of x back into the entry fee equation:\r<BR>\n" );
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document.write( "* Entry fee = 20 + 5x<BR>\n" );
document.write( "* Entry fee = 20 + 5(6)<BR>\n" );
document.write( "* Entry fee = 20 + 30<BR>\n" );
document.write( "* Entry fee = 50\r<BR>\n" );
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document.write( "**5. Calculate the Number of Players**\r<BR>\n" );
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document.write( "Substitute the value of x back into the number of players equation:\r<BR>\n" );
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document.write( "* Number of players = 80 - 5x<BR>\n" );
document.write( "* Number of players = 80 - 5(6)<BR>\n" );
document.write( "* Number of players = 80 - 30<BR>\n" );
document.write( "* Number of players = 50\r<BR>\n" );
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document.write( "**6. Calculate the Maximum Income**\r<BR>\n" );
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document.write( "* Maximum income = (Entry fee) * (Number of players)<BR>\n" );
document.write( "* Maximum income = 50 * 50<BR>\n" );
document.write( "* Maximum income = 2500\r<BR>\n" );
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document.write( "**Conclusion**\r<BR>\n" );
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document.write( "Elmer should charge PHP50 in order to maximize the income.<BR>\n" );
document.write( " </SPAN>\n" );
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