document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1141515>Question 1141515</A>: <I><SPAN STYLE=\"word-break: break-all;\"> It is given that the base and its two lateral side are 10cm in length. Find the measurement of the upper side so that the gutter can the hold the maximum amount of water. Give a reason to support your answer. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #762085 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13200)</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=762085\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "Let x be the angle the sides make with a perpendicular to the bases -- e.g., if the angles the sides make with the bottom base are 100 degrees, then x is 10 degrees.  Then<br><BR>\n" );
document.write( "the bottom base is 10<BR>\n" );
document.write( "the top base is 10+2*10sin(x)<BR>\n" );
document.write( "the height (depth) is 10cos(x)<br><BR>\n" );
document.write( "The maximum amount of water is when the cross sectional area is maximum.<br><BR>\n" );
document.write( "The area is height times the average of the two bases.<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A+=+10cos%28x%29%28%2810%2B%2810%2B20sin%28x%29%29%29%2F2%29 ALIGN=MIDDLE  ALT=\"A+=+10cos%28x%29%28%2810%2B%2810%2B20sin%28x%29%29%29%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A+=+100cos%28x%29%281%2Bsin%28x%29%29 ALIGN=MIDDLE  ALT=\"A+=+100cos%28x%29%281%2Bsin%28x%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "To find the maximum area, find where the derivative is zero.  Note we can ignore the constant 100 and find the maximum value of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28x%29%281%2Bsin%28x%29%29 ALIGN=MIDDLE  ALT=\"cos%28x%29%281%2Bsin%28x%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "By the product rule, the derivative is...<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28cos%28x%29%2Acos%28x%29%29%2B%28-sin%28x%29%29%281%2Bsin%28x%29%29 ALIGN=MIDDLE  ALT=\"%28cos%28x%29%2Acos%28x%29%29%2B%28-sin%28x%29%29%281%2Bsin%28x%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=cos%28x%29%5E2-sin%28x%29-sin%28x%29%5E2 ALIGN=MIDDLE  ALT=\"cos%28x%29%5E2-sin%28x%29-sin%28x%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Using <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28x%29%5E2-sin%28x%29%5E2+=+cos%282x%29%2C%3Cbr%3E%0D%0A%7B%7B%7Bcos%282x%29-sin%28x%29 ALIGN=MIDDLE  ALT=\"cos%28x%29%5E2-sin%28x%29%5E2+=+cos%282x%29%2C%3Cbr%3E%0D%0A%7B%7B%7Bcos%282x%29-sin%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "The derivative is zero when cos(2x) = sin(x).<br><BR>\n" );
document.write( "Knowing that the angle x is acute, and that for acute angles sin(x) = cos(90-x), we can determine that the derivative is zero when x is 30 degrees.<br><BR>\n" );
document.write( "This can be confirmed by graphing the area function on a graphing calculator and finding the value of x that produces the maximum area.<br><BR>\n" );
document.write( "ANSWER: The gutter will hold the maximum amount of water when the length of the top base is 10+20*sin(30) = 10+20(1/2) = 10+10 = 20.<BR>\n" );
document.write( " </SPAN>\n" );
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