document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=931525>Question 931525</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Please help me solve this problem: \r<BR>\n" );
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document.write( " Hiking Staff magazine says that the average hiker experiences a grade of 0.03 or less as flat. The Main Range Trail ascends eastbound out of Yakker's Lake according to H(x)= 0.75+(2/5)Log(x+1), where x is elapsed distance in miles eastbound from the lake, and H(x) is miles above the elevation of the lake. <BR>\n" );
document.write( " Mathematics Enthusiast magazine reports that the slope of form <BR>\n" );
document.write( " y=C+kLogb(x+1) is k/lnb*(1/x+1)for any positive value of x. In this context, \"Grade and \"Slope\" are synonyms. \r<BR>\n" );
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document.write( " Fred an average hiker, hikes eastbound out of Yakker's Lake. How far east of Yakker's Lake does Fred begin to think he is hiking on the flat?<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 #565608 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=565608\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The way I understand the problem, the Main Range Trail ascends eastbound out of Yakker's Lake according to<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=H%28x%29=+0.75%2B%282%2F5%29log%28%28x%2B1%29%29 ALIGN=MIDDLE  ALT=\"H%28x%29=+0.75%2B%282%2F5%29log%28%28x%2B1%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" > is elapsed distance in miles eastbound from the lake,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=H%28x%29 ALIGN=MIDDLE  ALT=\"H%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is miles above the elevation of the lake,<BR>\n" );
document.write( "and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28%28x%2B1%29%29 ALIGN=MIDDLE  ALT=\"log%28%28x%2B1%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%2810%2C%28x%2B1%29%29 ALIGN=MIDDLE  ALT=\"log%2810%2C%28x%2B1%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , the common logarithm (base 10 logarithm) .<BR>\n" );
document.write( "I also understand that Mathematics Enthusiast magazine reports that the slope of the graph for a function of the form<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=C%2Bk%2Alog%28b%2C%28x%2B1%29%29 ALIGN=MIDDLE  ALT=\"y=C%2Bk%2Alog%28b%2C%28x%2B1%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28k%2Fln%28b%29%29%2A%281%2F%28x%2B1%29%29 ALIGN=MIDDLE  ALT=\"%28k%2Fln%28b%29%29%2A%281%2F%28x%2B1%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > for any positive value of x.<BR>\n" );
document.write( " <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=H%28x%29=+0.75%2B%282%2F5%29log%2810%2C%28x%2B1%29%29 ALIGN=MIDDLE  ALT=\"H%28x%29=+0.75%2B%282%2F5%29log%2810%2C%28x%2B1%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is a function of the form <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=C%2Bk%2Alog%28b%2C%28x%2B1%29%29 ALIGN=MIDDLE  ALT=\"y=C%2Bk%2Alog%28b%2C%28x%2B1%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C=0.75 ALIGN=MIDDLE  ALT=\"C=0.75\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=10 ALIGN=MIDDLE  ALT=\"b=10\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=2%2F5=0.4 ALIGN=MIDDLE  ALT=\"k=2%2F5=0.4\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "so the slope of its graph is given by<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%280.4%2Fln%2810%29%29%2A%281%2F%28x%2B1%29%29 ALIGN=MIDDLE  ALT=\"%280.4%2Fln%2810%29%29%2A%281%2F%28x%2B1%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "So, the slope/grade of the trail east of Yakker's Lake is represented by the function<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=grade=0.4%2F%28ln%2810%29%2A%28x%2B1%29%29 ALIGN=MIDDLE  ALT=\"grade=0.4%2F%28ln%2810%29%2A%28x%2B1%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "Fred begin to think he is hiking on the flat when <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=grade=0.03 ALIGN=MIDDLE  ALT=\"grade=0.03\"   DISABLED_style= \"max-width:90%; height:auto;\" > , so our equation is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.03=0.4%2F%28ln%2810%29%2A%28x%2B1%29%29 ALIGN=MIDDLE  ALT=\"0.03=0.4%2F%28ln%2810%29%2A%28x%2B1%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Solving:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.03=0.4%2F%28ln%2810%29%2A%28x%2B1%29%29 ALIGN=MIDDLE  ALT=\"0.03=0.4%2F%28ln%2810%29%2A%28x%2B1%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.03%2A%28x%2B1%29=0.4%2Fln%2810%29 ALIGN=MIDDLE  ALT=\"0.03%2A%28x%2B1%29=0.4%2Fln%2810%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B1=0.4%2F%28ln%2810%29%2A0.03%29 ALIGN=MIDDLE  ALT=\"x%2B1=0.4%2F%28ln%2810%29%2A0.03%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=0.4%2F%28ln%2810%29%2A0.03%29-1 ALIGN=MIDDLE  ALT=\"x=0.4%2F%28ln%2810%29%2A0.03%29-1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Using the approximate value <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ln%2810%29=2.3026 ALIGN=MIDDLE  ALT=\"ln%2810%29=2.3026\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "(or using however many digits my calculator can handle),<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=4.79 ALIGN=MIDDLE  ALT=\"x=4.79\"   DISABLED_style= \"max-width:90%; height:auto;\" > (rounded).<BR>\n" );
document.write( "So, Fred starts thinking the trail is flat when he is 4.8 miles east of the lake. </SPAN>\n" );
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