document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=4627>Question 4627</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A rancher wants to yield the maximum amount of edible beef off his square mile of land. At first he finds that as he adds more cattle, his yield goes up. However, if he overgrazes, the harvest goes down. An agricultural agent tells him that his yield will vary quadratically with the number of animals that he grazes. For five head of cattle, his production is 8,750 pounds of beef, and for 10 head, he reaps 15,000 pounds of beef. He has no production when he was grazing no cattle.<BR>\n" );
document.write( "a) Define the variables, write the odered pairs, and find the particular equation of this function.<BR>\n" );
document.write( "b) What number of cattle will give him his maximum profit in number of pounds?<BR>\n" );
document.write( "c) What is this weight?<BR>\n" );
document.write( "d) What is the domain and range of this function?<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 #2163 by </B><A HREF=/tutors/aboutme.mpl?userid=rapaljer><B>rapaljer(4671)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=rapaljer><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=rapaljer><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=2163\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> I think we need Longjonsilver or Kenny for this one, but I'll give it a try.  First I had never heard of varying quadratically before, but I'm assuming that it means <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=Y+=+ax%5E2+%2B+bx+%2B+c+ ALIGN=MIDDLE  ALT=\"Y+=+ax%5E2+%2B+bx+%2B+c+\"   DISABLED_style= \"max-width:90%; height:auto;\" >, where x is the number of animals, and Y is the Yield in pounds of beef.\r<BR>\n" );
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document.write( "Three pieces of information are given in the problem to allow you to solve for the constants a, b, and c.\r<BR>\n" );
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document.write( "Y(5) = 8750<BR>\n" );
document.write( "Y(10) = 15000<BR>\n" );
document.write( "Y(0) = 0\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=Y+=+ax%5E2+%2B+bx+%2B+c+ ALIGN=MIDDLE  ALT=\"Y+=+ax%5E2+%2B+bx+%2B+c+\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <BR>\n" );
document.write( "Substituting into this equation YIELDS [Pun!!  Get it??]:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8750+=+a%285%29%5E2+%2B+b%285%29+%2B+c ALIGN=MIDDLE  ALT=\"8750+=+a%285%29%5E2+%2B+b%285%29+%2B+c\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=15000+=+a%2810%29%5E2+%2B+b%2810%29+%2B+c+ ALIGN=MIDDLE  ALT=\"15000+=+a%2810%29%5E2+%2B+b%2810%29+%2B+c+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+0+=+a%280%29%5E2+%2B+b%280%29+%2B+c+ ALIGN=MIDDLE  ALT=\"+0+=+a%280%29%5E2+%2B+b%280%29+%2B+c+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "8500 = 25a + 5b<BR>\n" );
document.write( "15000= 100a + 10b<BR>\n" );
document.write( "0 = 0(a) + 0(b) + c<BR>\n" );
document.write( "The last equation gives you immediately that c = 0.\r<BR>\n" );
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document.write( "Multiply the first equation by -2:<BR>\n" );
document.write( "-17500 = -50a - 10b<BR>\n" );
document.write( "15000 = 100a + 10b\r<BR>\n" );
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document.write( "-2500 = 50a\r<BR>\n" );
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document.write( "a=-50\r<BR>\n" );
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document.write( "Substituting back into either equation gives you b = 2000\r<BR>\n" );
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document.write( "The equation of Yield is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=Y+=+-50x%5E2+%2B+2000x+ ALIGN=MIDDLE  ALT=\"Y+=+-50x%5E2+%2B+2000x+\"   DISABLED_style= \"max-width:90%; height:auto;\" >, which is a parabolic graph opening down.\r<BR>\n" );
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document.write( "The maximum yield occurs at the vertex, which can be found at <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+-b%2F2a ALIGN=MIDDLE  ALT=\"x+=+-b%2F2a\"   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%2F%282%2A%28-50%29%29+=+-2000%2F%28-100%29+=+20 ALIGN=MIDDLE  ALT=\"x+=+-2000%2F%282%2A%28-50%29%29+=+-2000%2F%28-100%29+=+20\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=Y%2820%29+=+-50%2A20%5E2+%2B+2000%2A20+=+-20000+%2B+40000+=+20000 ALIGN=MIDDLE  ALT=\"Y%2820%29+=+-50%2A20%5E2+%2B+2000%2A20+=+-20000+%2B+40000+=+20000\"   DISABLED_style= \"max-width:90%; height:auto;\" > = maximum yield\r<BR>\n" );
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document.write( "Domain of this problem is from x = 0 to 40:  [0, 40]<BR>\n" );
document.write( "Range Y is from 0 to the maximum at 20,000:  [0, 20,000]\r<BR>\n" );
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document.write( "R^2 at SCC\r<BR>\n" );
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