document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=85996>Question 85996</A>: <I><SPAN STYLE=\"word-break: break-all;\"> This is a problem I have run into here at work. I need to solve for x in the following equation. Thanks for the help.<BR>\n" );
document.write( "S=(1440*T)/(N*x*pi^2*D^2*acos((D-x)/D)) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #62194 by </B><A HREF=/tutors/aboutme.mpl?userid=bucky><B>bucky(2189)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=bucky><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=62194\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> I assume that by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=acos%28%28D-x%29%2FD%29 ALIGN=MIDDLE  ALT=\"acos%28%28D-x%29%2FD%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > you mean <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=arccosine%28%28D-x%29%2FD%29 ALIGN=MIDDLE  ALT=\"arccosine%28%28D-x%29%2FD%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or “the angle whose cosine is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28D-x%29%2FD ALIGN=MIDDLE  ALT=\"%28D-x%29%2FD\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( ".<BR>\n" );
document.write( "Here’s some food for thought.<BR>\n" );
document.write( ".<BR>\n" );
document.write( "You have:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S+=+%281440%2AT%29%2F%28N%2A%28pi%5E2%29%2A%28D%5E2%29%2Ax%2Aacos%28%28D-x%29%2FD%29%29 ALIGN=MIDDLE  ALT=\"S+=+%281440%2AT%29%2F%28N%2A%28pi%5E2%29%2A%28D%5E2%29%2Ax%2Aacos%28%28D-x%29%2FD%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "Let’s begin by getting all the terms that contain x on the left side and everything <BR>\n" );
document.write( "else on the right side. The first step is to multiply both sides by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2Aacos%28%28D-x%29%2FD%29 ALIGN=MIDDLE  ALT=\"x%2Aacos%28%28D-x%29%2FD%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. <BR>\n" );
document.write( "When you do, the equation becomes:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S%2Ax%2Aacos%28%28D-x%29%2FD%29+=++%281440%2AT%29%2F%28N%2A%28pi%5E2%29%2A%28D%5E2%29%29 ALIGN=MIDDLE  ALT=\"S%2Ax%2Aacos%28%28D-x%29%2FD%29+=++%281440%2AT%29%2F%28N%2A%28pi%5E2%29%2A%28D%5E2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "Next divide both sides by S and get:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2Aacos%28%28D-x%29%2FD%29+=++%281440%2AT%29%2F%28N%2A%28pi%5E2%29%2A%28D%5E2%29%2AS%29 ALIGN=MIDDLE  ALT=\"x%2Aacos%28%28D-x%29%2FD%29+=++%281440%2AT%29%2F%28N%2A%28pi%5E2%29%2A%28D%5E2%29%2AS%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "Now, for reasons that will become apparent later, let’s multiply both sides of this equation <BR>\n" );
document.write( "by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D%5E2 ALIGN=MIDDLE  ALT=\"D%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >. <BR>\n" );
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document.write( "This makes the equation become:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D%5E2%2Ax%2Aacos%28%28D-x%29%2FD%29+=+%281440%2AT%29%2F%28N%2A%28pi%5E2%29%2AS%29 ALIGN=MIDDLE  ALT=\"D%5E2%2Ax%2Aacos%28%28D-x%29%2FD%29+=+%281440%2AT%29%2F%28N%2A%28pi%5E2%29%2AS%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "Now let’s suppose we can force D to be some value other than zero (because if it were 0, <BR>\n" );
document.write( "then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28D-x%29%2FD ALIGN=MIDDLE  ALT=\"%28D-x%29%2FD\"   DISABLED_style= \"max-width:90%; height:auto;\" > would involve a division by zero). Suppose for example we could make <BR>\n" );
document.write( "D equal to x.  If we could do this, then substituting x for D would make the equation become:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2Ax%2Aacos%28%28x-x%29%2Fx%29+=+%281440%2AT%29%2F%28N%2A%28pi%5E2%29%2AS%29 ALIGN=MIDDLE  ALT=\"x%5E2%2Ax%2Aacos%28%28x-x%29%2Fx%29+=+%281440%2AT%29%2F%28N%2A%28pi%5E2%29%2AS%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "Notice two things: first <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2Ax+=+x%5E3 ALIGN=MIDDLE  ALT=\"x%5E2%2Ax+=+x%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > and second <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=acos%28%28x-x%29%2Fx%29+=+acos%280%2Fx%29+=+acos%280%29 ALIGN=MIDDLE  ALT=\"acos%28%28x-x%29%2Fx%29+=+acos%280%2Fx%29+=+acos%280%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "If you limit the angle to being between <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0 ALIGN=MIDDLE  ALT=\"0\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Api ALIGN=MIDDLE  ALT=\"2%2Api\"   DISABLED_style= \"max-width:90%; height:auto;\" >, then there are two possible <BR>\n" );
document.write( "values for this angle when D = x.  Those values for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=acos%280%29 ALIGN=MIDDLE  ALT=\"acos%280%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > are <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi%2F2 ALIGN=MIDDLE  ALT=\"pi%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283%2Api%29%2F2 ALIGN=MIDDLE  ALT=\"%283%2Api%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  <BR>\n" );
document.write( ".<BR>\n" );
document.write( "Let’s substitute <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi%2F2 ALIGN=MIDDLE  ALT=\"pi%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=acos%280%29 ALIGN=MIDDLE  ALT=\"acos%280%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. This changes the equation to:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E3%29%2A%28pi%2F2%29+=+%281440%2AT%29%2F%28N%2A%28pi%5E2%29%2AS%29 ALIGN=MIDDLE  ALT=\"%28x%5E3%29%2A%28pi%2F2%29+=+%281440%2AT%29%2F%28N%2A%28pi%5E2%29%2AS%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "Next divide both sides of the equation by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi%2F2 ALIGN=MIDDLE  ALT=\"pi%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > or multiply by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Fpi ALIGN=MIDDLE  ALT=\"2%2Fpi\"   DISABLED_style= \"max-width:90%; height:auto;\" > which leads to:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3+=%282%2Fpi%29%2A%28%281440%2AT%29%2F%28N%2A%28pi%5E2%29%2AS%29%29 ALIGN=MIDDLE  ALT=\"x%5E3+=%282%2Fpi%29%2A%28%281440%2AT%29%2F%28N%2A%28pi%5E2%29%2AS%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "And the right side multiplies out to:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3+=+%282880%2AT%29%2F%28N%2A%28pi%5E3%29%2AS%29 ALIGN=MIDDLE  ALT=\"x%5E3+=+%282880%2AT%29%2F%28N%2A%28pi%5E3%29%2AS%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "Then you can solve for x by taking the cube root of both sides.<BR>\n" );
document.write( ".<BR>\n" );
document.write( "But don’t forget that this requires that you be able to make D equal x.<BR>\n" );
document.write( ".<BR>\n" );
document.write( "And don’t forget that you have another value possible because <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=acos%280%29+=+%283%2Api%29%2F2 ALIGN=MIDDLE  ALT=\"acos%280%29+=+%283%2Api%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "is also a possibility.<BR>\n" );
document.write( ".<BR>\n" );
document.write( "Let’s now substitute <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283%2Api%29%2F2 ALIGN=MIDDLE  ALT=\"%283%2Api%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=acos%280%29 ALIGN=MIDDLE  ALT=\"acos%280%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. This changes the equation to:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3%2A%283%2Api%29%2F2+=+%281440%2AT%29%2F%28N%2Api%5E2%2AS%29 ALIGN=MIDDLE  ALT=\"x%5E3%2A%283%2Api%29%2F2+=+%281440%2AT%29%2F%28N%2Api%5E2%2AS%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "Next divide both sides of the equation by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283%2Api%29%2F2 ALIGN=MIDDLE  ALT=\"%283%2Api%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > [or multiply both sides by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2F%283%2Api%29 ALIGN=MIDDLE  ALT=\"2%2F%283%2Api%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >] <BR>\n" );
document.write( "which leads to:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3+=+%282%2F%283%2Api%29%29%2A%28%281440%2AT%29%2F%28N%2A%28pi%5E2%29%2AS%29%29 ALIGN=MIDDLE  ALT=\"x%5E3+=+%282%2F%283%2Api%29%29%2A%28%281440%2AT%29%2F%28N%2A%28pi%5E2%29%2AS%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "And the right side multiplies out to:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3+=+%28%282880%2AT%29%29%2F%283%2A%28N%2A%28pi%5E3%29%2AS%29%29 ALIGN=MIDDLE  ALT=\"x%5E3+=+%28%282880%2AT%29%29%2F%283%2A%28N%2A%28pi%5E3%29%2AS%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "Again, you can solve for x by taking the cube root of both sides.<BR>\n" );
document.write( ".<BR>\n" );
document.write( "You can do a similar analysis for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=acos%28-1%29 ALIGN=MIDDLE  ALT=\"acos%28-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > which implies that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28D-x%29%2FD+=+-1 ALIGN=MIDDLE  ALT=\"%28D-x%29%2FD+=+-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > which, <BR>\n" );
document.write( "when solved for D, says that D must be set so that it equals <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2F2 ALIGN=MIDDLE  ALT=\"x%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So this leads to <BR>\n" );
document.write( "replacing <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=D%5E2+ ALIGN=MIDDLE  ALT=\"D%5E2+\"   DISABLED_style= \"max-width:90%; height:auto;\" > with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2F2%29%5E2 ALIGN=MIDDLE  ALT=\"%28x%2F2%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > which is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E2%29%2F4 ALIGN=MIDDLE  ALT=\"%28x%5E2%29%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > and replacing <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=acos%28-1%29 ALIGN=MIDDLE  ALT=\"acos%28-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "with the angle <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pi ALIGN=MIDDLE  ALT=\"pi\"   DISABLED_style= \"max-width:90%; height:auto;\" > because that is the only angle between <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0 ALIGN=MIDDLE  ALT=\"0\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Api ALIGN=MIDDLE  ALT=\"2%2Api\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "that has as its cosine the value -1.  This leads to:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E2%29%2F4+%2A+x+%2A%28pi%29+=+%281440%2AT%29%2F%28N%2A%28pi%5E2%29%2AS%29 ALIGN=MIDDLE  ALT=\"%28x%5E2%29%2F4+%2A+x+%2A%28pi%29+=+%281440%2AT%29%2F%28N%2A%28pi%5E2%29%2AS%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "Next divide both sides of the equation by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28pi%2F4%29 ALIGN=MIDDLE  ALT=\"%28pi%2F4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > [or multiply both sides by <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F%284%2Api%29 ALIGN=MIDDLE  ALT=\"1%2F%284%2Api%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >] which leads to:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E3%29%2F4+=+%281%2F%284%2Api%29%29%2A%28%281440%2AT%29%2F%28N%2A%28pi%5E2%29%2AS%29%29 ALIGN=MIDDLE  ALT=\"%28x%5E3%29%2F4+=+%281%2F%284%2Api%29%29%2A%28%281440%2AT%29%2F%28N%2A%28pi%5E2%29%2AS%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "And the right side multiplies out to:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3+=+%28%281440%2AT%29%29%2F%284%2A%28N%2A%28pi%5E3%29%2AS%29%29 ALIGN=MIDDLE  ALT=\"x%5E3+=+%28%281440%2AT%29%29%2F%284%2A%28N%2A%28pi%5E3%29%2AS%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( ".<BR>\n" );
document.write( "Again, you can solve for x by taking the cube root of both sides.<BR>\n" );
document.write( ".<BR>\n" );
document.write( "I won’t belabor the point, but you can continue this analysis for other values. But it is <BR>\n" );
document.write( "all based on the fact that to do this you must be able to control the value of D.  <BR>\n" );
document.write( ".<BR>\n" );
document.write( "Like I said at the beginning, just food for thought. Hope the process stimulates you to find <BR>\n" );
document.write( "something that you might be able to do to solve for x.  I’ve done this rather rapidly, <BR>\n" );
document.write( "so be sure to check the above to make sure that I didn’t make some dumb math mistake … <BR>\n" );
document.write( "which I often do unless I’m on my 4th cup of coffee … and I’m not at present.<BR>\n" );
document.write( ".\r<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );