document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1157615>Question 1157615</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I just do not understand optimization every problem confuses me please explain this problem and its solution.\r<BR>\n" );
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document.write( "An island is 2mi due north of its closest point along a straight shoreline. A visitor is staying at a cabin on the shore that is 6mi west of that point. The visitor is planning to go from the cabin to the island. Suppose the visitor runs at a rate of 8mph and swims at a rate of 3mph. How far should the visitor run before swimming to minimize the time it takes to reach the island? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #780512 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=780512\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Times New Roman\" size=\"+2\">\r<BR>\n" );
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document.write( "In this problem, we are given distances and rates and we are looking to minimize time.  We know that distance = rate times time, so time is distance divided by rate.\r<BR>\n" );
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document.write( "Here is the particular situation graphically:\r<BR>\n" );
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document.write( " <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/display-illustration.mpl?tutor=solver91311&illustration_name=Figure1\">\r<BR>\n" );
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document.write( "The total time is the sum of the Running time and the Swimming time, so\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ T(x)\ =\ \frac{6\,-\,x}{8}\ +\ \frac{sqrt{x^2\,+\,4}}{3}\">\r<BR>\n" );
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document.write( "And note that the domain of the function for this particular problem is <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \{x\,\in\,\mathbb{R}\,|\,0\,\leq\,x\,\leq\,6\}\">\r<BR>\n" );
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document.write( "Figure 2 is the graph of the time function:\r<BR>\n" );
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document.write( " <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/display-illustration.mpl?tutor=solver91311&illustration_name=Figure_2\">\r<BR>\n" );
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document.write( "Note that the graph has a minimum value somewhere in the neighborhood of <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\ =\ 0.8\"> (the vertical grid lines are spaced 0.5 apart)\r<BR>\n" );
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document.write( "The exact point is where the slope of the tangent line to the curve is zero.\r<BR>\n" );
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document.write( "Calculate:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \frac{d}{dx}\(\frac{6\,-\,x}{8}\ +\ \frac{sqrt{x^2\,+\,4}}{3}\)\">\r<BR>\n" );
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document.write( "Then set the derivative function equal to zero and solve for <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\">.  The answer to the question as posed is then <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large 6\,-\,x\">\r<BR>\n" );
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document.write( "John<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e^{i\pi}\ +\ 1\ =\ 0\"><BR>\n" );
document.write( "My calculator said it, I believe it, that settles it<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src=\"http://c0rk.blogs.com/gr0undzer0/darwin-fish.jpg\"><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \ \\r\"> </SPAN>\n" );
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