document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1151077>Question 1151077</A>: <I><SPAN STYLE=\"word-break: break-all;\"> flag poles A and B are both perpendicular to the ground. Wires from the top of one flagpole are connected to the bottom of the other flagpole such that the point where the wires intersect is 10 feet above the ground. If the height of both A and B are an integer number of feet, how many possible heights are there for A? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #772698 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=772698\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "The basic problem here is a common one.  With the two poles of heights a and b, the height above the ground at which the two wires cross is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ab%2F%28a%2Bb%29 ALIGN=MIDDLE  ALT=\"ab%2F%28a%2Bb%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "So in this problem we have<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ab%2F%28a%2Bb%29=10 ALIGN=MIDDLE  ALT=\"ab%2F%28a%2Bb%29=10\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "and we need to determine the number of possible values for a if a and b are both integers.<br><BR>\n" );
document.write( "This is a Diophantine equation -- we have two variables but only one equation; but the number of solutions is limited by the requirement that both variables be integers.<br><BR>\n" );
document.write( "The following is one standard way for solving such equations.<br><BR>\n" );
document.write( "(1) Solve the equation for one variable in terms of the other.<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ab%2F%28a%2Bb%29+=+10 ALIGN=MIDDLE  ALT=\"ab%2F%28a%2Bb%29+=+10\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ab+=+10a%2B10b ALIGN=MIDDLE  ALT=\"ab+=+10a%2B10b\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ab-10b+=+10a ALIGN=MIDDLE  ALT=\"ab-10b+=+10a\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%28a-10%29+=+10a ALIGN=MIDDLE  ALT=\"b%28a-10%29+=+10a\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b+=+10a%2F%28a-10%29 ALIGN=MIDDLE  ALT=\"b+=+10a%2F%28a-10%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b+=+%2810a-100%2B100%29%2F%28a-10%29 ALIGN=MIDDLE  ALT=\"b+=+%2810a-100%2B100%29%2F%28a-10%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b+=+%2810a-100%29%2F%28a-10%29%2B100%2F%28a-10%29 ALIGN=MIDDLE  ALT=\"b+=+%2810a-100%29%2F%28a-10%29%2B100%2F%28a-10%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b+=+10%2B100%2Fa-10 ALIGN=MIDDLE  ALT=\"b+=+10%2B100%2Fa-10\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "(2) b and 10 are integers; that means <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=100%2F%28a-10%29 ALIGN=MIDDLE  ALT=\"100%2F%28a-10%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > must be an integer.  And that means (a-10) must be a factor of 100.<br><BR>\n" );
document.write( "We could identify all the possible values of a; but the problem only asks for the number of possible values for a.  100 has 9 positive factors, so there are 9 possible values for a.<br><BR>\n" );
document.write( "ANSWER:  There are 9 possible heights for pole A.<br><BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );