document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1039917>Question 1039917</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Suppose we have a graph of parametric equations x=bsin(t), y=b(cos(t))^3 on where t is on [0,2pi]. <BR>\n" );
document.write( "A. Find the point(s) on the given graph where the slope of the tangent line is 3/4. (Find t values not x,y values)<BR>\n" );
document.write( "B. Determine the area, in terms of b, of the region enclosed by the graph.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #654725 by </B><A HREF=/tutors/aboutme.mpl?userid=robertb><B>robertb(5830)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=robertb><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=robertb><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=654725\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> A. Find the point(s) on the given graph where the slope of the tangent line is 3/4. (Find t values not x,y values).\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=bsint ALIGN=MIDDLE  ALT=\"x=bsint\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=b%28cost%29%5E3 ALIGN=MIDDLE  ALT=\"y=b%28cost%29%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "==> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dx%2Fdt+=+b%2Acost ALIGN=MIDDLE  ALT=\"dx%2Fdt+=+b%2Acost\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dy%2Fdt+=+-3b%28cost%29%5E2%2Asint ALIGN=MIDDLE  ALT=\"dy%2Fdt+=+-3b%28cost%29%5E2%2Asint\"   DISABLED_style= \"max-width:90%; height:auto;\" > after differentiating both x and y wrt t.\r<BR>\n" );
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document.write( "==> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dy%2Fdx+=+-3sint%2Acost ALIGN=MIDDLE  ALT=\"dy%2Fdx+=+-3sint%2Acost\"   DISABLED_style= \"max-width:90%; height:auto;\" > after division.\r<BR>\n" );
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document.write( "To find the point(s) on the graph where the slope of the tangent line is 3/4, let <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dy%2Fdx+=+-3sint%2Acost+=+3%2F4 ALIGN=MIDDLE  ALT=\"dy%2Fdx+=+-3sint%2Acost+=+3%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > .\r<BR>\n" );
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document.write( "==> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-sint%2Acost+=+1%2F4 ALIGN=MIDDLE  ALT=\"-sint%2Acost+=+1%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" >  ==>  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2sint%2Acost+=+1%2F2 ALIGN=MIDDLE  ALT=\"-2sint%2Acost+=+1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > ==> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin2t+=+-1%2F2 ALIGN=MIDDLE  ALT=\"sin2t+=+-1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "==> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2t+=+%287%2Api%29%2F6 ALIGN=MIDDLE  ALT=\"2t+=+%287%2Api%29%2F6\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2t+=+%2811%2Api%29%2F6 ALIGN=MIDDLE  ALT=\"2t+=+%2811%2Api%29%2F6\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "==> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t+=+%287%2Api%29%2F12 ALIGN=MIDDLE  ALT=\"t+=+%287%2Api%29%2F12\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t+=+%2811%2Api%29%2F12 ALIGN=MIDDLE  ALT=\"t+=+%2811%2Api%29%2F12\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "B. Determine the area, in terms of b, of the region enclosed by the graph. <BR>\n" );
document.write( "To determine the area, first express y in terms of x.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=bsint ALIGN=MIDDLE  ALT=\"x=bsint\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=b%28cost%29%5E3+ ALIGN=MIDDLE  ALT=\"y=b%28cost%29%5E3+\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "==> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2Fb=sint ALIGN=MIDDLE  ALT=\"x%2Fb=sint\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28y%2Fb%29%5E%281%2F3%29=cost+ ALIGN=MIDDLE  ALT=\"%28y%2Fb%29%5E%281%2F3%29=cost+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "==> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2Fb%5E2+%2B+y%5E%282%2F3%29%2Fb%5E%282%2F3%29+=+1 ALIGN=MIDDLE  ALT=\"x%5E2%2Fb%5E2+%2B+y%5E%282%2F3%29%2Fb%5E%282%2F3%29+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "==> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5E%282%2F3%29%2Fb%5E%282%2F3%29+=+1+-x%5E2%2Fb%5E2++=+%28b%5E2+-+x%5E2%29%2Fb%5E2 ALIGN=MIDDLE  ALT=\"y%5E%282%2F3%29%2Fb%5E%282%2F3%29+=+1+-x%5E2%2Fb%5E2++=+%28b%5E2+-+x%5E2%29%2Fb%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "==>  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5E%282%2F3%29=%28b%5E%282%2F3%29%2Fb%5E2%29%2A%28b%5E2+-+x%5E2%29 ALIGN=MIDDLE  ALT=\"y%5E%282%2F3%29=%28b%5E%282%2F3%29%2Fb%5E2%29%2A%28b%5E2+-+x%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "==> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5E%282%2F3%29=%281%2Fb%5E%284%2F3%29%29%2A%28b%5E2+-+x%5E2%29 ALIGN=MIDDLE  ALT=\"y%5E%282%2F3%29=%281%2Fb%5E%284%2F3%29%29%2A%28b%5E2+-+x%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "==> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%281%2Fb%5E2%29%2A%28b%5E2+-+x%5E2%29%5E%283%2F2%29 ALIGN=MIDDLE  ALT=\"y=%281%2Fb%5E2%29%2A%28b%5E2+-+x%5E2%29%5E%283%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28-1%2Fb%5E2%29%2A%28b%5E2+-+x%5E2%29%5E%283%2F2%29 ALIGN=MIDDLE  ALT=\"y=%28-1%2Fb%5E2%29%2A%28b%5E2+-+x%5E2%29%5E%283%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Now for purpose of getting the area assume that b > 0. (The sense of the graph in this case is clockwise.  b < 0 would only change the direction to ccw but would yield the same graph and the same area.)\r<BR>\n" );
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document.write( "==> Area = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%284%2Fb%5E2%29int%28%28b%5E2+-+x%5E2%29%5E%283%2F2%29%2C+dx%2C+0%2Cb%29 ALIGN=MIDDLE  ALT=\"%284%2Fb%5E2%29int%28%28b%5E2+-+x%5E2%29%5E%283%2F2%29%2C+dx%2C+0%2Cb%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "(I wrote it this way rather than <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282%2Fb%5E2%29int%28%28b%5E2+-+x%5E2%29%5E%283%2F2%29%2C+dx%2C+-b%2Cb%29 ALIGN=MIDDLE  ALT=\"%282%2Fb%5E2%29int%28%28b%5E2+-+x%5E2%29%5E%283%2F2%29%2C+dx%2C+-b%2Cb%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  They give the same answer.)\r<BR>\n" );
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document.write( "Let x = bsint.<BR>\n" );
document.write( "==> dx = bcostdt\r<BR>\n" );
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document.write( "==> Area =  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%284%2Fb%5E2%29int%28%28b%5E2+-+b%5E2%2A%28sint%29%5E2%29%5E%283%2F2%29%2C+bcostdt%2C+0%2Cpi%2F2%29+=+%284%2Fb%5E2%29int%28b%5E4%281+-%28sint%29%5E2%29%5E%283%2F2%29%2C+costdt%2C+0%2Cpi%2F2%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
document.write( "\n" );
document.write( "=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4b%5E2%2Aint%28%28%28cost%29%5E2%29%5E%283%2F2%29%2C+costdt%2C+0%2Cpi%2F2%29+=4b%5E2%2Aint%28%28cost%29%5E4%2C+dt%2C+0%2Cpi%2F2%29+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
document.write( "\n" );
document.write( "Using <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28cost%29%5E2+=+%281%2Bcos2t%29%2F2 ALIGN=MIDDLE  ALT=\"%28cost%29%5E2+=+%281%2Bcos2t%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >,\r<BR>\n" );
document.write( "\n" );
document.write( "Area = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2%2Aint%28%281%2B2cos2t%2B%28cos2t%29%5E2%29%2C+dt%2C+0%2Cpi%2F2%29+ ALIGN=MIDDLE  ALT=\"b%5E2%2Aint%28%281%2B2cos2t%2B%28cos2t%29%5E2%29%2C+dt%2C+0%2Cpi%2F2%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
document.write( "\n" );
document.write( "=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2%2Aint%28%281%2B2cos2t%2B%281%2Bcos4t%29%2F2%29%2C+dt%2C+0%2Cpi%2F2%29+ ALIGN=MIDDLE  ALT=\"b%5E2%2Aint%28%281%2B2cos2t%2B%281%2Bcos4t%29%2F2%29%2C+dt%2C+0%2Cpi%2F2%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
document.write( "\n" );
document.write( "=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2%2Aint%28%283%2F2%2B2cos2t%2B%281%2F2%29%2Acos4t%29%2C+dt%2C+0%2Cpi%2F2%29+ ALIGN=MIDDLE  ALT=\"b%5E2%2Aint%28%283%2F2%2B2cos2t%2B%281%2F2%29%2Acos4t%29%2C+dt%2C+0%2Cpi%2F2%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
document.write( "\n" );
document.write( "=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2%2A%28%283%2F2%29t%2B+sin2t+%2B+%281%2F8%29%2Asin4t%29%5B0%5D%5E%28pi%2F2%29 ALIGN=MIDDLE  ALT=\"b%5E2%2A%28%283%2F2%29t%2B+sin2t+%2B+%281%2F8%29%2Asin4t%29%5B0%5D%5E%28pi%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
document.write( "\n" );
document.write( "=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E2%2A%283%2F2%29%2A%28pi%2F2%29+=+%283b%5E2%2Api%29%2F4 ALIGN=MIDDLE  ALT=\"b%5E2%2A%283%2F2%29%2A%28pi%2F2%29+=+%283b%5E2%2Api%29%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );