document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=201400>Question 201400</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hello! I need a help ASAP!!! <BR>\n" );
document.write( "Problem 3.<BR>\n" );
document.write( "In 1920, the record for a certain race was 46.4 sec. In 1990, it was 45.0 sec.<BR>\n" );
document.write( "Let R(t)= the record in the race and t= the number of the year since 1920.\r<BR>\n" );
document.write( "\n" );
document.write( "a) Find a linear function that the date<BR>\n" );
document.write( "b) Use the function in (a) to predict the record in 2003 and in 2006.<BR>\n" );
document.write( "c) Find the year when the record will be 44.56 sec\r<BR>\n" );
document.write( "\n" );
document.write( "Find a linear function that fits the data.<BR>\n" );
document.write( "R(t)= ?? <BR>\n" );
document.write( "What is the predicted record for 2003? <BR>\n" );
document.write( "What is the predicted record for 2006? <BR>\n" );
document.write( "In what year will the predicted record be 44.56 sec??<BR>\n" );
document.write( "Thank you so much for your help!!!\r<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 #151627 by </B><A HREF=/tutors/aboutme.mpl?userid=jsmallt9><B>jsmallt9(3758)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jsmallt9><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=151627\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> In general when you want to find the equation of a line, you need the slope of the line and at least one point on the line. To start with you have two points:<BR>\n" );
document.write( "<PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">\r\n" );
document.write( "Year   t (Year-1920)   R(t)   Point\r\n" );
document.write( "1920   0               46.4   (0, 46.4)\r\n" );
document.write( "1990   70              45.0   (70, 45.0)\r\n" );
document.write( "</PRE><BR>\n" );
document.write( "We have the point(s) we need but we do not have the slope. With the two points we can calculate the slope using the slope formula:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m+=+%28y%5B2%5D-y%5B1%5D%29%2F%28x%5B2%5D-x%5B1%5D%29 ALIGN=MIDDLE  ALT=\"m+=+%28y%5B2%5D-y%5B1%5D%29%2F%28x%5B2%5D-x%5B1%5D%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Substituting the coordinates of the two points above in the formula:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m+=+%2846.4+-+45.0%29%2F%280+-+70%29 ALIGN=MIDDLE  ALT=\"m+=+%2846.4+-+45.0%29%2F%280+-+70%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Subtracting we get<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m+=+%281.4%29%2F%28-70%29 ALIGN=MIDDLE  ALT=\"m+=+%281.4%29%2F%28-70%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Dividing 1.4 by -70 we get<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m+=+-0.02 ALIGN=MIDDLE  ALT=\"m+=+-0.02\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Now we have the slope and at least one point. There are a couple of ways to find the equation of the line:<ul><li>Using the Point-Slope formula: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+-+y%5B1%5D+=+m%28x+-+x%5B1%5D%29 ALIGN=MIDDLE  ALT=\"y+-+y%5B1%5D+=+m%28x+-+x%5B1%5D%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ></li><li>Using the Slope-Intercept form: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+mx+%2B+b ALIGN=MIDDLE  ALT=\"y+=+mx+%2B+b\"   DISABLED_style= \"max-width:90%; height:auto;\" ></li></ul>Since most students seem to the prefer the second method that is how we will proceed.<BR>\n" );
document.write( "First we substitute the slope and the coordinates of a point (it doesn't matter which point as long as it is on the line) into the Slope-Intercept form.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+mx+%2B+b ALIGN=MIDDLE  ALT=\"y+=+mx+%2B+b\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Substituting the coordinates of (0, 46.4) in for the x and the y and substituting -0.02 in for the m we get<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2846.4%29+=+%28-0.02%29%2A%280%29+%2B+b ALIGN=MIDDLE  ALT=\"%2846.4%29+=+%28-0.02%29%2A%280%29+%2B+b\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "We can now solve for b. Since -0.02 * 0 is zero and since 0 + b is b we get<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=46.4+=+b ALIGN=MIDDLE  ALT=\"46.4+=+b\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Now that we have m (-0.02) and b (46.4) we can write the equation of our line. In doing so we will use t instead of x and R(t) instead of y.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R%28t%29+=+-0.02t+%2B+46.4 ALIGN=MIDDLE  ALT=\"R%28t%29+=+-0.02t+%2B+46.4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "which is the answer to part (a).<br><BR>\n" );
document.write( "For part (b), to calculate the record in 2003 we will use 83 for t (since 2003-1920 = 83) and figure out R(83). To calculate the record in 2006 we will use t = 86 and calculate R(86).<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R%2883%29+=+-0.02%2883%29+%2B+46.4 ALIGN=MIDDLE  ALT=\"R%2883%29+=+-0.02%2883%29+%2B+46.4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Multiplying -0.02 by 83 we get<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R%2883%29+=+-1.66+%2B+46.4 ALIGN=MIDDLE  ALT=\"R%2883%29+=+-1.66+%2B+46.4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Adding we get<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R%2883%29+=+44.74 ALIGN=MIDDLE  ALT=\"R%2883%29+=+44.74\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Similarly<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R%2886%29+=+-0.02%2886%29+%2B+46.4 ALIGN=MIDDLE  ALT=\"R%2886%29+=+-0.02%2886%29+%2B+46.4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Simplifying like above<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=R%2886%29+=+44.68 ALIGN=MIDDLE  ALT=\"R%2886%29+=+44.68\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "For part (c) we will use R(t) = 44.56 and solve for t:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=44.56+=+-0.02t+%2B+46.4 ALIGN=MIDDLE  ALT=\"44.56+=+-0.02t+%2B+46.4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Subtracting 46.4 from both sides we get<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1.84+=+-0.02t ALIGN=MIDDLE  ALT=\"-1.84+=+-0.02t\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Dividing both sides by -0.02 we get<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=92+=+t ALIGN=MIDDLE  ALT=\"92+=+t\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Since t = 92 the year must be 92+1920 or 2012\r<BR>\n" );
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