document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=855600>Question 855600</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The base of a right triangle is decreasing at a rate of 2 inches per minute, and the height of the triangle is increasing at a rate of 4 inches per minute. When the base of the triangle is 2 feet and the height is 3 feet, how fast is the area of the triangle changing in square inches per minute?  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #515457 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=515457\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The way I see it, this is a calculus problem.<BR>\n" );
document.write( "With <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t ALIGN=MIDDLE  ALT=\"t\"   DISABLED_style= \"max-width:90%; height:auto;\" >= time (in minutes) counted from the moment the base of the triangle is 2 feet and the height is 3 feet,<BR>\n" );
document.write( "the base (in inches), height (in inches), and area (in square inches) are all functions of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t ALIGN=MIDDLE  ALT=\"t\"   DISABLED_style= \"max-width:90%; height:auto;\" > :<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=base=2%2A12-2t=24-2t ALIGN=MIDDLE  ALT=\"base=2%2A12-2t=24-2t\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=height=3%2A12%2B4t=36%2B4t ALIGN=MIDDLE  ALT=\"height=3%2A12%2B4t=36%2B4t\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=Area=base%2Aheight%2F2=%2824-2t%29%2836%2B4t%29%2F2=%28864-72t%2B96t-8t%5E2%29%2F2=%28864%2B24t-8t%5E2%29%2F2=432%2B12t-4t%5E2 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The rate of change of the area (in square inches per minute) is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dArea%2Fdt=12-8t ALIGN=MIDDLE  ALT=\"dArea%2Fdt=12-8t\"   DISABLED_style= \"max-width:90%; height:auto;\" > which changes with time.<BR>\n" );
document.write( "At <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=0 ALIGN=MIDDLE  ALT=\"t=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > (the moment the base of the triangle is 2 feet and the height is 3 feet)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dArea%2Fdt=12-8%2A1=highlight%2812%29 ALIGN=MIDDLE  ALT=\"dArea%2Fdt=12-8%2A1=highlight%2812%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " <BR>\n" );
document.write( "Without invoking calculus, the only solution I see is saying that the rate is the slope of the tangent to the graph.<BR>\n" );
document.write( "With <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=t ALIGN=MIDDLE  ALT=\"x=t\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=Area-432=12t-4t%5E2=12x-4x%5E2 ALIGN=MIDDLE  ALT=\"y=Area-432=12t-4t%5E2=12x-4x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > the graph would be<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28300%2C300%2C-5%2C5%2C-25%2C25%2C12x-4x%5E2%2C12x%29 ALIGN=MIDDLE  ALT=\"graph%28300%2C300%2C-5%2C5%2C-25%2C25%2C12x-4x%5E2%2C12x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The tangent <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=kx ALIGN=MIDDLE  ALT=\"y=kx\"   DISABLED_style= \"max-width:90%; height:auto;\" > , with slope <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k ALIGN=MIDDLE  ALT=\"k\"   DISABLED_style= \"max-width:90%; height:auto;\" > passes through the origin<BR>\n" );
document.write( "and intersects <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=12x-4x%5E2 ALIGN=MIDDLE  ALT=\"y=12x-4x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > at only one point.<BR>\n" );
document.write( "That means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=kx=12x-4x%5E2 ALIGN=MIDDLE  ALT=\"kx=12x-4x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > must have only one solution.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=kx=12x-4x%5E2 ALIGN=MIDDLE  ALT=\"kx=12x-4x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x%5E2-12x%2Bkx=0 ALIGN=MIDDLE  ALT=\"4x%5E2-12x%2Bkx=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x%5E2%2B%28k-12%29x=0 ALIGN=MIDDLE  ALT=\"4x%5E2%2B%28k-12%29x=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%284x%2Bk-12%29=0 ALIGN=MIDDLE  ALT=\"x%284x%2Bk-12%29=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "has the solutions <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=0 ALIGN=MIDDLE  ALT=\"x=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%2812-k%29%2F4 ALIGN=MIDDLE  ALT=\"x=%2812-k%29%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Those solutions are the same only when <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12-k=0 ALIGN=MIDDLE  ALT=\"12-k=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=k=12 ALIGN=MIDDLE  ALT=\"k=12\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "which makes the line tangent.<BR>\n" );
document.write( "A line with another slope would intersect the graph at two different points, with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=0 ALIGN=MIDDLE  ALT=\"x=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%2812-k%29%2F4 ALIGN=MIDDLE  ALT=\"x=%2812-k%29%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > . </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );