document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=758089>Question 758089</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A small island is 4 miles from the nearest point P on the straight shoreline of a large lake. If a woman on the island can row a boat 2 miles per hour and can walk 3 miles per hour, where should the boat be landed in order to arrive at a town 12 miles down the shore from P in the least time? Let x be the distance between point P and where the boat lands on the lake shore. \r<BR>\n" );
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document.write( "(A) Enter a function T(x) that describes the total amount of time the trip takes as a function of the distance x. <BR>\n" );
document.write( "T(x) =  \r<BR>\n" );
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document.write( "(B) What is the distance x=c that minimizes the travel time? Note: The answer to this problem requires that you enter the correct units. <BR>\n" );
document.write( "c =  . \r<BR>\n" );
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document.write( "(C) What is the least travel time? Note: The answer to this problem requires that you enter the correct units. <BR>\n" );
document.write( "The least travel time is  . \r<BR>\n" );
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document.write( "(D) Recall that the second derivative test says that if T&#8242;(c)=0 and T&#8242;&#8242;(c)>0, then T has a local minimum at c. What is T&#8242;&#8242;(c)? <BR>\n" );
document.write( "T&#8242;&#8242;(c) =  <BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #461262 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=461262\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The distance walked would be (12-x) miles.<BR>\n" );
document.write( "The distance rowed would be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%2816%2Bx%5E2%29 ALIGN=MIDDLE  ALT=\"sqrt%2816%2Bx%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > miles.<BR>\n" );
document.write( "(A) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28T%28x%29=sqrt%2816%2Bx%5E2%29%2F2%2B%2812-x%29%2F3%29 ALIGN=MIDDLE  ALT=\"highlight%28T%28x%29=sqrt%2816%2Bx%5E2%29%2F2%2B%2812-x%29%2F3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , measured in hours.<BR>\n" );
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document.write( "(B) T'(x) = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F2%29%281%2F2%2Fsqrt%2816%2Bx%5E2%29%29%2A2x-1%2F3=x%2F2%2Fsqrt%2816%2Bx%5E2%29-1%2F3 ALIGN=MIDDLE  ALT=\"%281%2F2%29%281%2F2%2Fsqrt%2816%2Bx%5E2%29%29%2A2x-1%2F3=x%2F2%2Fsqrt%2816%2Bx%5E2%29-1%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > is zero for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=c ALIGN=MIDDLE  ALT=\"x=c\"   DISABLED_style= \"max-width:90%; height:auto;\" > so<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c%2F2sqrt%2816%2Bc%5E2%29-1%2F3=0 ALIGN=MIDDLE  ALT=\"c%2F2sqrt%2816%2Bc%5E2%29-1%2F3=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c%2F2sqrt%2816%2Bc%5E2%29=1%2F3 ALIGN=MIDDLE  ALT=\"c%2F2sqrt%2816%2Bc%5E2%29=1%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c%2Fsqrt%2816%2Bc%5E2%29=2%2F3 ALIGN=MIDDLE  ALT=\"c%2Fsqrt%2816%2Bc%5E2%29=2%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c%5E2%2F%2816%2Bc%5E2%29=4%2F9 ALIGN=MIDDLE  ALT=\"c%5E2%2F%2816%2Bc%5E2%29=4%2F9\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9c%5E2=4%2816%2Bc%5E2%29 ALIGN=MIDDLE  ALT=\"9c%5E2=4%2816%2Bc%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9c%5E2=64%2B4c%5E2 ALIGN=MIDDLE  ALT=\"9c%5E2=64%2B4c%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5c%5E2=64 ALIGN=MIDDLE  ALT=\"5c%5E2=64\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c%5E2=64%2F5 ALIGN=MIDDLE  ALT=\"c%5E2=64%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c=sqrt%2864%2F5%29=8%2Fsqrt%285%29=8sqrt%285%29%2F5 ALIGN=MIDDLE  ALT=\"c=sqrt%2864%2F5%29=8%2Fsqrt%285%29=8sqrt%285%29%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The approximate value is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28c=3.5777miles%29 ALIGN=MIDDLE  ALT=\"highlight%28c=3.5777miles%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Since you are entering an answer, the expected answer may be \"3.6 miles\", but I would not know what format will be accepted. Maybe \"miles\" must be abreviated. Maybe spaces are not accepted, and the computer wants \"c=3.58mi\" for an answer. Maybe an answer starting with \"c=\" is not accepted.<BR>\n" );
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document.write( "(C) The least travel time is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=T%28c%29=sqrt%2816%2Bc%5E2%29%2F2%2B%2812-c%29%2F3%29 ALIGN=MIDDLE  ALT=\"T%28c%29=sqrt%2816%2Bc%5E2%29%2F2%2B%2812-c%29%2F3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "I calculated <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=T%28c%29=approx5.49hours ALIGN=MIDDLE  ALT=\"T%28c%29=approx5.49hours\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "(D) T\"(x) = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F2%2Fsqrt%2816%2Bx%5E2%29-x%5E2%2F2%2F%2816%2Bx%5E2%29%5E%283%2F2%29 ALIGN=MIDDLE  ALT=\"1%2F2%2Fsqrt%2816%2Bx%5E2%29-x%5E2%2F2%2F%2816%2Bx%5E2%29%5E%283%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8%2F%2816%2Bx%5E2%29%5E%283%2F2%29 ALIGN=MIDDLE  ALT=\"8%2F%2816%2Bx%5E2%29%5E%283%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "I calculated T\"(c) = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=approx0.052hour%2Fmile%5E2 ALIGN=MIDDLE  ALT=\"approx0.052hour%2Fmile%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "I see no reason to make us calculate that. Finding that T(c)=0, while T(x)<0 for x < c and T(x)>0 for x > c proves that T(x) decreases for x < c and increases for x > c, so there is a minimum at x=c. </SPAN>\n" );
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