document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1183764>Question 1183764</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Let (x,y) be a point on the triangular region bounded by the line 3x + 4y = 12 and the coordinate axes.  Determine the points (x,y) in this region which  give the minimum and maximum sums of distances of the point from the line 3x + 4y = 12 and from the coordinate axes. Please include explanation.  Thank you! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #814244 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13203)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=814244\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "The response from the other tutor contains errors, which I am sure she will find and correct after seeing that I have provided a response to your question.<br><BR>\n" );
document.write( "The given triangle has vertices (0,0), (4,0), and (0,3).<br><BR>\n" );
document.write( "Let the point P (x,y) be in that triangle.<br><BR>\n" );
document.write( "The distance of P from the x-axis is y; the distance from the y-axis is x.<br><BR>\n" );
document.write( "The distance of P from the line 3x+4y=12, or 3x+4y-12=0, is<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28%283x%2B4y-12%29%2Fsqrt%283%5E2%2B4%5E2%29%29=abs%28%283x%2B4y-12%29%2F5%29 ALIGN=MIDDLE  ALT=\"abs%28%283x%2B4y-12%29%2Fsqrt%283%5E2%2B4%5E2%29%29=abs%28%283x%2B4y-12%29%2F5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "For all points inside the triangle,<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283x%2B4y-12%29%2F5%29 ALIGN=MIDDLE  ALT=\"%283x%2B4y-12%29%2F5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "is negative.  So<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28%283x%2B4y-12%29%2F5%29=%2812-3x-4y%29%2F5 ALIGN=MIDDLE  ALT=\"abs%28%283x%2B4y-12%29%2F5%29=%2812-3x-4y%29%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "and the sum of the distances of P from the three sides of the triangle is<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2By%2B%2812-3x-4y%29%2F5=%2812%2B2x%2By%29%2F5 ALIGN=MIDDLE  ALT=\"x%2By%2B%2812-3x-4y%29%2F5=%2812%2B2x%2By%29%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Given that expression for the sum of the distances of P from the three sides of the triangle, it is clear that the minimum sum is at (0,0), where the sum of the distances is 12/5.<br><BR>\n" );
document.write( "For the maximum sum, note that for a given value of x the sum of the distances is greatest is when y is as large as possible; and for a given value of y the sum of the distances is greatest is when x is as large as possible.  That means the maximum sum is when the point P is somewhere on the boundary line 3x+4y=12.<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%2B4y=12 ALIGN=MIDDLE  ALT=\"3x%2B4y=12\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4y=-3x%2B12 ALIGN=MIDDLE  ALT=\"4y=-3x%2B12\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28-3%2F4%29x%2B3 ALIGN=MIDDLE  ALT=\"y=%28-3%2F4%29x%2B3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "The sum of the distance is then<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2812%2B2x%2B%28%28-3%2F4%29x%2B3%29%29%2F5=%28%285%2F4%29x%2B15%29%2F5=%281%2F4%29x%2B3 ALIGN=MIDDLE  ALT=\"%2812%2B2x%2B%28%28-3%2F4%29x%2B3%29%29%2F5=%28%285%2F4%29x%2B15%29%2F5=%281%2F4%29x%2B3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "And clearly the maximum value of that expression is when x is as large as possible -- at (4,0).<br><BR>\n" );
document.write( "The maximum sum is then<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2812%2B2x%2By%29%2F5=%2812%2B8%29%2F5=20%2F5=4 ALIGN=MIDDLE  ALT=\"%2812%2B2x%2By%29%2F5=%2812%2B8%29%2F5=20%2F5=4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "ANSWERS:<BR>\n" );
document.write( "minimum sum 12/5 = 2.4, at (0,0)<BR>\n" );
document.write( "maximum sum 4, at (4,0)<br><BR>\n" );
document.write( " </SPAN>\n" );
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