document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1011898>Question 1011898</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A sheet of metal is 60 cm wide and 10 m long. It is bent along its width to form a gutter with a cross section that is an isosceles trapezoid with 120 degree angles. \r<BR>\n" );
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document.write( "a. Express the volume V of the gutter as a function of x, the length in cm of one of the equal sides. ( hint: Volume= area of trapezoid x length of gutter)\r<BR>\n" );
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document.write( "B. For what value of x is the volume of the gutter a maximum? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #627772 by </B><A HREF=/tutors/aboutme.mpl?userid=ankor@dixie-net.com><B>ankor@dixie-net.com(22740)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=ankor@dixie-net.com><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=ankor@dixie-net.com><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=627772\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> A sheet of metal is 60 cm wide and 10 m long. It is bent along its width to form a gutter with a cross section that is an isosceles trapezoid with 120 degree angles.<BR>\n" );
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document.write( "It will help to draw this out. Label the height of the gutter h, sides of the gutter as x<BR>\n" );
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document.write( "a. Express the volume V of the gutter as a function of x, the length in cm of one of the equal sides. ( hint: Volume= area of trapezoid x length of gutter)<BR>\n" );
document.write( "The area of a trapezoid: A = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28h%2F2%29%28a%2Bb%29 ALIGN=MIDDLE  ALT=\"%28h%2F2%29%28a%2Bb%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Ahh yes, sorry! Can I blame it on New year's eve wine? No? Ok let me try to straighten this out.<BR>\n" );
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document.write( "  x\__/x, is a picture of the end of the gutter<BR>\n" );
document.write( "We have to find a, b and h in terms of x<BR>\n" );
document.write( "b = (60-2x), the shortest width of the gutter<BR>\n" );
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document.write( "a = the longer width of the gutter: this width is the shorter width plus the sides of the right triangle formed on both sides.<BR>\n" );
document.write( " the angle of this right triangle: 120-90 = 30 degrees.<BR>\n" );
document.write( " Find the length of this side (s) in terms of x<BR>\n" );
document.write( "sin(30) = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=s%2Fx ALIGN=MIDDLE  ALT=\"s%2Fx\"   DISABLED_style= \"max-width:90%; height:auto;\" >, the side opposite<BR>\n" );
document.write( "s = .5x<BR>\n" );
document.write( "a = (60-2x) + 2(s)<BR>\n" );
document.write( "replace s with .5x<BR>\n" );
document.write( "a = 60 - 2x + 2(.5x)<BR>\n" );
document.write( "a = 60 - 2x + x<BR>\n" );
document.write( "a = (60 - x) is the longer side of the trapezoid<BR>\n" );
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document.write( "h = the height of the gutter is the adjacent side of this same right triangle<BR>\n" );
document.write( "cos(30) = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=h%2Fx ALIGN=MIDDLE  ALT=\"h%2Fx\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "h = .866x<BR>\n" );
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document.write( "Substituting in A = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28h%2F2%29%28a%2Bb%29 ALIGN=MIDDLE  ALT=\"%28h%2F2%29%28a%2Bb%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "A = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28.866x%2F2%29%2860-x%29+%2B+%2860-2x%29%29 ALIGN=MIDDLE  ALT=\"%28.866x%2F2%29%2860-x%29+%2B+%2860-2x%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "A = .433x(120 - 3x)<BR>\n" );
document.write( "A = 52x - 1.3x^2 sq cm is the area of the end of the gutter<BR>\n" );
document.write( "To find the volume multiply this by 1000 cm (10 meters)<BR>\n" );
document.write( "V = 1000(52x - 1.3x^2) <BR>\n" );
document.write( "V = 52000x - 1300x^2 cu/cm is volume of the gutter<BR>\n" );
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document.write( "B. For what value of x is the volume of the gutter a maximum? <BR>\n" );
document.write( "The above is a quadratic equation.<BR>\n" );
document.write( "the max occurs at the axis of symmetry which is x = -b/(2a)<BR>\n" );
document.write( "x = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-52000%29%2F%282%2A-1300%29 ALIGN=MIDDLE  ALT=\"%28-52000%29%2F%282%2A-1300%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "x = 20 cm  is the sides on the gutter for max volume<BR>\n" );
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document.write( "Let me know if you get this OK. ankor@att.net </SPAN>\n" );
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