document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1203691>Question 1203691</A>: <I><SPAN STYLE=\"word-break: break-all;\"> For a certain model of car the distance d required to stop the vehicle if it is traveling at v mi/h is given by the formula d = v + v2 20 where d is measured in feet. kerry wants her stopping distance not to exceed 40ft. at what range of speeds (in mi/h) can she travel? (enter your answer using interval notation </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #839462 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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=839462\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "The formula <font color=blue>d = v + v2 20</font> seems strangely formed. Please double-check to make sure you typed this correctly.\r<BR>\n" );
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document.write( "After some online searching, I found a similar problem with the formula <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d+=+v+%2B+%28v%5E2%29%2F20 ALIGN=MIDDLE  ALT=\"d+=+v+%2B+%28v%5E2%29%2F20\"   DISABLED_style= \"max-width:90%; height:auto;\" > and the stopping distance was 240 feet (instead of 40 feet)<BR>\n" );
document.write( "If this is the formula and stopping distance your teacher gave you, then read on. If not then let me know.\r<BR>\n" );
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document.write( "d = stopping distance or braking distance = the distance the car travels when the brakes are first applied to when the car comes to a complete stop\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d+%3C=+240 ALIGN=MIDDLE  ALT=\"d+%3C=+240\"   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=v+%2B+%28v%5E2%29%2F20+%3C=+240 ALIGN=MIDDLE  ALT=\"v+%2B+%28v%5E2%29%2F20+%3C=+240\"   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=20%2A%28v+%2B+%28v%5E2%29%2F20%29+%3C=+20%2A240 ALIGN=MIDDLE  ALT=\"20%2A%28v+%2B+%28v%5E2%29%2F20%29+%3C=+20%2A240\"   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=20v+%2B+v%5E2+%3C=+4800 ALIGN=MIDDLE  ALT=\"20v+%2B+v%5E2+%3C=+4800\"   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=20v+%2B+v%5E2-4800+%3C=+0 ALIGN=MIDDLE  ALT=\"20v+%2B+v%5E2-4800+%3C=+0\"   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=v%5E2+%2B+20v+-+4800+%3C=+0 ALIGN=MIDDLE  ALT=\"v%5E2+%2B+20v+-+4800+%3C=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Let's complete the square<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=v%5E2+%2B+20v+%2B+0+-+4800+%3C=+0 ALIGN=MIDDLE  ALT=\"v%5E2+%2B+20v+%2B+0+-+4800+%3C=+0\"   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=v%5E2+%2B+20v+%2B+100+-+100+-+4800+%3C=+0 ALIGN=MIDDLE  ALT=\"v%5E2+%2B+20v+%2B+100+-+100+-+4800+%3C=+0\"   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=%28v%5E2+%2B+20v+%2B+100%29+-+100+-+4800+%3C=+0 ALIGN=MIDDLE  ALT=\"%28v%5E2+%2B+20v+%2B+100%29+-+100+-+4800+%3C=+0\"   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=%28v%2B10%29%5E2+-+100+-+4800+%3C=+0 ALIGN=MIDDLE  ALT=\"%28v%2B10%29%5E2+-+100+-+4800+%3C=+0\"   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=%28v%2B10%29%5E2+-+4900+%3C=+0 ALIGN=MIDDLE  ALT=\"%28v%2B10%29%5E2+-+4900+%3C=+0\"   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=%28v%2B10%29%5E2+%3C=+4900 ALIGN=MIDDLE  ALT=\"%28v%2B10%29%5E2+%3C=+4900\"   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=sqrt%28%28v%2B10%29%5E2%29+%3C=+sqrt%284900%29 ALIGN=MIDDLE  ALT=\"sqrt%28%28v%2B10%29%5E2%29+%3C=+sqrt%284900%29\"   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=abs%28v%2B10%29+%3C=+70 ALIGN=MIDDLE  ALT=\"abs%28v%2B10%29+%3C=+70\"   DISABLED_style= \"max-width:90%; height:auto;\" > Use the rule sqrt(x^2) = |x| \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-70+%3C=+v%2B10+%3C=+70 ALIGN=MIDDLE  ALT=\"-70+%3C=+v%2B10+%3C=+70\"   DISABLED_style= \"max-width:90%; height:auto;\" >  we use the rule that |x| < k becomes -k < x < k, for some positive value k.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-70-10+%3C=+v%2B10-10+%3C=+70-10 ALIGN=MIDDLE  ALT=\"-70-10+%3C=+v%2B10-10+%3C=+70-10\"   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=-80+%3C=+v+%3C=+60 ALIGN=MIDDLE  ALT=\"-80+%3C=+v+%3C=+60\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Another way to solve would be to factor <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=v%5E2+%2B+20v+-+4800+%3C=+0 ALIGN=MIDDLE  ALT=\"v%5E2+%2B+20v+-+4800+%3C=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28v%2B80%29%28v-60%29+%3C=+0 ALIGN=MIDDLE  ALT=\"%28v%2B80%29%28v-60%29+%3C=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The roots are v = -80 and v = 60<BR>\n" );
document.write( "The region between these roots will make the left hand side either 0 or negative.\r<BR>\n" );
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document.write( "Clearly we cannot have negative speeds.<BR>\n" );
document.write( "Therefore we must revise <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-80+%3C=+v+%3C=+60 ALIGN=MIDDLE  ALT=\"-80+%3C=+v+%3C=+60\"   DISABLED_style= \"max-width:90%; height:auto;\" > to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0+%3C=+v+%3C=+60 ALIGN=MIDDLE  ALT=\"0+%3C=+v+%3C=+60\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "To be more technical, since she is driving, her speed cannot be zero. <BR>\n" );
document.write( "So we further would say <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0+%3C+v+%3C=+60 ALIGN=MIDDLE  ALT=\"0+%3C+v+%3C=+60\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "That converts to the interval notation (0, 60]<BR>\n" );
document.write( "The parenthesis excludes the endpoint 0.<BR>\n" );
document.write( "A square bracket is used to include the endpoint 60.\r<BR>\n" );
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document.write( "Answer: <font color=red size=4>(0, 60]</font><BR>\n" );
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