document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1200565>Question 1200565</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The operator of a hotel estimates that 500 rooms per night will be rented if the room per night is $100. For each $10 increase in the price of room, six fewer rooms per night be rented. Determine a linear function that will predicted the number of rooms that will be rented per night for a given price room. Use this model to predict the number of rooms that will be rented if the room rate is $150/night. <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 #834719 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=834719\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> i believe your simplified function will be y = -.6x + 560.\r<BR>\n" );
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document.write( "i derived it as follows:\r<BR>\n" );
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document.write( "start with two points.<BR>\n" );
document.write( "the first point is (100,500).<BR>\n" );
document.write( "the second point is (110,494).\r<BR>\n" );
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document.write( "from this, you can create a linear function of y = mx + b, where m is the slope and b is the y-intercept.\r<BR>\n" );
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document.write( "the formula for the slope is m = (y2 - y1) / (x2 - x1)\r<BR>\n" );
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document.write( "label the first point as (x1,y1) = (100,500).<BR>\n" );
document.write( "label the second point as (x2,y2) = (110,494).\r<BR>\n" );
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document.write( "m = (y2 - y1) / (x2 - x1) m = (494 - 500) / (110 - 100).<BR>\n" );
document.write( "simplify to get:<BR>\n" );
document.write( "m = -6/10 = -.6\r<BR>\n" );
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document.write( "that's your slope.\r<BR>\n" );
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document.write( "when x = 100, you want y to be 500).<BR>\n" );
document.write( "the point representing this becomes (100,500).<BR>\n" );
document.write( "you can use this point to find the y-intercept.<BR>\n" );
document.write( "replace y with 500 and x with 100 in the equation to get:<BR>\n" );
document.write( "y = -.6x + b becomes:<BR>\n" );
document.write( "500 = -.6 * 100 + b<BR>\n" );
document.write( "simplify to get:<BR>\n" );
document.write( "500 = -60 + b<BR>\n" );
document.write( "add 60 to both sides of the equation to get:<BR>\n" );
document.write( "560 = b<BR>\n" );
document.write( "that's your y-intercept.\r<BR>\n" );
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document.write( "the equation becomes:<BR>\n" );
document.write( "y = -.6x + 560\r<BR>\n" );
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document.write( "i graphed the eqwuation to show you that the equation is accurate.<BR>\n" );
document.write( "the graph is shown below.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2023/022402.jpg\">\r<BR>\n" );
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document.write( "x is the price of the room for one night.<BR>\n" );
document.write( "y is the number of rooms rented.\r<BR>\n" );
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document.write( "when x = 100, y = 500, which is as stated in the problem statement.<BR>\n" );
document.write( "when x = 110, y = 494, which indicates that, as the price goes up 10 dollars, the number of rooms rented is 6 less.\r<BR>\n" );
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document.write( "this continues for all values of x and corresponding values of y.\r<BR>\n" );
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document.write( "the equation can be used to determine the number of rooms rented when the price is 150.<BR>\n" );
document.write( "as seen on the graph, the number of rooms rented will be 470 when the price is 150.<BR>\n" );
document.write( "the price went up by 5 * 10 = 50 and the number of rooms rented went down by 5 * 6 = 30.<BR>\n" );
document.write( "you can derive this from the equation as follows:<BR>\n" );
document.write( "the equation is y = -.6 * x + 560<BR>\n" );
document.write( "when x = 150, the equation becomes y = -.6 * 150 + 560.<BR>\n" );
document.write( "simplify to get y = 470.<BR>\n" );
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