document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=998474>Question 998474</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Every Week that a farmer waits, his potato crop increases 100 bushels, but the price drop 10 cents a bushel. If he digs potatoes today, he would have 600 bushels worth $1 per bushel. Show that he should dig and sell his potatoes at the end of two weeks.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #616270 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=616270\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> this is a little tricky, but you can figure it out once you know how.\r<BR>\n" );
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document.write( "we'll let x = the number of weeks.\r<BR>\n" );
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document.write( "when x = 0, the farmer has 600 bushels to sell.<BR>\n" );
document.write( "when x = 1, the farmer has 700 bushels to sell.<BR>\n" );
document.write( "each week, the farmer has 100 more bushels to sell.\r<BR>\n" );
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document.write( "the formula for that would be b = 100 * x + 600\r<BR>\n" );
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document.write( "b is the number of bushels.<BR>\n" );
document.write( "x is the number of weeks.\r<BR>\n" );
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document.write( "if you apply the formula, you can see that:\r<BR>\n" );
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document.write( "when x = 0, b = 600<BR>\n" );
document.write( "when x = 1, b = 700<BR>\n" );
document.write( "when x = 2, b = 800<BR>\n" );
document.write( "etc.\r<BR>\n" );
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document.write( "now the price that he can get for each bushels is 1.00 today and .90 in a week and .80 in another week.\r<BR>\n" );
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document.write( "the formula for that would be p = (1 - .10 * x) * b\r<BR>\n" );
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document.write( "p is the price that he gets for the potatoes in week x.\r<BR>\n" );
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document.write( "when x = 0, p = 1 * b<BR>\n" );
document.write( "when x = 1, p = (1-.10*1) * b = .90 * b<BR>\n" );
document.write( "when x = 2, p = (1 - .10*2) * b = .80 * b<BR>\n" );
document.write( "etc.\r<BR>\n" );
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document.write( "so you have two formulas:\r<BR>\n" );
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document.write( "the first formuls is b = 100 * x + 600\r<BR>\n" );
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document.write( "the second formula is p = (1 - .10 * x) * b\r<BR>\n" );
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document.write( "i will combine these formula later, but for now, i wish to show you how you would analyze this with just the information you already  have.\r<BR>\n" );
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document.write( "we'll do an analysis for x = 0, 1, 2, 3, and 4.\r<BR>\n" );
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document.write( "since p depends on b, we'll evaluate for b first, and then for p.\r<BR>\n" );
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document.write( "you'll get a table that looks like this:\r<BR>\n" );
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document.write( "                x               b               p\r\n" );
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document.write( "                0               600             600\r\n" );
document.write( "                1               700             630\r\n" );
document.write( "                2               800             640\r\n" );
document.write( "                3               900             630\r\n" );
document.write( "                4               1000            600\r\n" );
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document.write( "from the table, you can see that the price the farmer gets for the potatoes will peak at the end of 2 weeks.\r<BR>\n" );
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document.write( "an example of how we got the number for b and p is shown below:\r<BR>\n" );
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document.write( "when x = 2, b = 100 * x + 600 which becomes:<BR>\n" );
document.write( "b = 100 * 2 + 600 which becomes:<BR>\n" );
document.write( "b = 200 + 600 which becomes:<BR>\n" );
document.write( "b = 800.\r<BR>\n" );
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document.write( "when x = 2:<BR>\n" );
document.write( "p = (1-.10*x) * b<BR>\n" );
document.write( "since b = 800 when x = 2, the formula becomes:<BR>\n" );
document.write( "p = (1 - .10*2)*800 which becomes:<BR>\n" );
document.write( "p = .8 * 800 which becomes:<BR>\n" );
document.write( "p = 640.\r<BR>\n" );
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document.write( "in fact, what you see is the values given by a quadratic equation.\r<BR>\n" );
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document.write( "how did we get that quadratic equation?\r<BR>\n" );
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document.write( "look below for the derivation of the quadratic equation and the graph of that equation.\r<BR>\n" );
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document.write( "go back to where we derived both formulas:\r<BR>\n" );
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document.write( "the first formuls is b = 100 * x + 600\r<BR>\n" );
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document.write( "the second formula is p = (1 - .10 * x) * b\r<BR>\n" );
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document.write( "the second formula depends on the value of b from the first formula.\r<BR>\n" );
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document.write( "we can replace b in the second formula with the value of b from the first formula.\r<BR>\n" );
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document.write( "we will get:\r<BR>\n" );
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document.write( "p = (1 - .10 * x) * b becomes:\r<BR>\n" );
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document.write( "p = (1 - .10 * x) * (100 * x + 600)\r<BR>\n" );
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document.write( "b was replaced with its equivalent value of (100 * x + 600)\r<BR>\n" );
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document.write( "we can graph this formula as is by replacing p with y to get:\r<BR>\n" );
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document.write( "y = (1 - .10 * x) * (100 * x + 600)\r<BR>\n" );
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document.write( "we can also convert it to ax^2 + bx + c form by performing the multiplication indicated.\r<BR>\n" );
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document.write( "when we perform the multiplication indicated, we get:\r<BR>\n" );
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document.write( "y = (1 - .10 * x) * (100 * x + 600) becomes:<BR>\n" );
document.write( "y = (1 * 100 * x) + (1 * 600) - (.10 * x * 100 * x) - (.10 * 600 * x)<BR>\n" );
document.write( "we simplify this to get:<BR>\n" );
document.write( "y = 100x + 600 - 10x^2 - 60x<BR>\n" );
document.write( "we combine like terms to get:<BR>\n" );
document.write( "y = 40x + 600 - 10x^2<BR>\n" );
document.write( "we rearrange the expression on the right in descending order of the power of each term to get:<BR>\n" );
document.write( "y = -10x^2 + 40x + 600\r<BR>\n" );
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document.write( "the two equations of:\r<BR>\n" );
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document.write( "[y = (1 - .10 * x) * (100 * x + 600)] and [y = -10x^2 + 40x + 600] are equivalent.\r<BR>\n" );
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document.write( "the graph of both equations will be identical.\r<BR>\n" );
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document.write( "this means if you show both equations on the same graph, their graphs will superimpose on each other and look like one graph rather than 2.\r<BR>\n" );
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document.write( "the graph of both of these equations is shown below:\r<BR>\n" );
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document.write( "the black line is the graph of both of these equations.<BR>\n" );
document.write( "the orange line shows the value of x = 2.<BR>\n" );
document.write( "the intersection point of the black line and the orange line shows the value of y when the value of x is equal to 2.<BR>\n" );
document.write( "this intersection point agrees with the table.<BR>\n" );
document.write( "the value of y is 640 when the value of x = 2.\r<BR>\n" );
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document.write( "this means the price the farmer gets for the potatoes is max at the end of 2 weeks and is equal to 640 dollars.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2015/102301.jpg\" alt=\"$$$\" </>\r<BR>\n" );
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document.write( "you can also find the max point by using the formula that tells you:\r<BR>\n" );
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document.write( "x value of max/min point of a quadratic equation is:\r<BR>\n" );
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document.write( "x = -b/2a\r<BR>\n" );
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document.write( "in the equation of y = -10x^2 + 40x + 600:\r<BR>\n" );
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document.write( "a = -10<BR>\n" );
document.write( "b = 40<BR>\n" );
document.write( "c = 600\r<BR>\n" );
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document.write( "x value of max/min point is -40 / -20 = 2.<BR>\n" );
document.write( "y value when x value is 2 is equal to -10*2^2 + 40*2 + 600.<BR>\n" );
document.write( "that becomes -40 + 80 + 600 which becomes:<BR>\n" );
document.write( "40 + 600 which becomes:<BR>\n" );
document.write( "y = 640 when x = 2.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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