document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1201107>Question 1201107</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A surveyor standing W 25 degrees S of a tower measures the angle of elevation of the top of the tower as 40 degrees 30. From a position E 23 degrees S from the tower the l elevation of the top is 37 degrees 15. Determine the height of the tower if the distance between the two observations is 75m. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #835362 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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=835362\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Notation like 40°30' is read out as \"40 degrees 30 minutes\".<BR>\n" );
document.write( "\"minutes\" refers specifically to <a  rel=nofollow HREF= \"https://en.wikipedia.org/wiki/Minute_and_second_of_arc\">arc minutes</a>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "There are 60 arc minutes in a full degree.<BR>\n" );
document.write( "30 arc minutes is 30/60 = 1/2 = 0.5 of a degree.<BR>\n" );
document.write( "15 arc minutes is 15/60 = 1/4 = 0.25 of a degree.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Therefore,<BR>\n" );
document.write( "40 degrees 30 minutes = 40.5 degrees <BR>\n" );
document.write( "37 degrees 15 minutes = 37.15 degrees <BR>\n" );
document.write( "which represent the angles of elevation.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "This is what it looks like from a birds eye view<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%0D%0Adrawing%28400%2C400%2C-6%2C7%2C-5%2C5%2C%0D%0Ablue%28%0D%0Aline%28-5%2C0%2C-4.75%2C0%29%2Cline%28-4.5%2C0%2C-4.25%2C0%29%2Cline%28-4%2C0%2C-3.75%2C0%29%2Cline%28-3.5%2C0%2C-3.25%2C0%29%2Cline%28-3%2C0%2C-2.75%2C0%29%2Cline%28-2.5%2C0%2C-2.25%2C0%29%2Cline%28-2%2C0%2C-1.75%2C0%29%2Cline%28-1.5%2C0%2C-1.25%2C0%29%2Cline%28-1%2C0%2C-0.75%2C0%29%2Cline%28-0.5%2C0%2C-0.25%2C0%29%2Cline%280%2C0%2C0.25%2C0%29%2Cline%280.5%2C0%2C0.75%2C0%29%2Cline%281%2C0%2C1.25%2C0%29%2Cline%281.5%2C0%2C1.75%2C0%29%2Cline%282%2C0%2C2.25%2C0%29%2Cline%282.5%2C0%2C2.75%2C0%29%2Cline%283%2C0%2C3.25%2C0%29%2Cline%283.5%2C0%2C3.75%2C0%29%2Cline%284%2C0%2C4.25%2C0%29%2Cline%284.5%2C0%2C4.75%2C0%29%0D%0A%29%2C%0D%0A%0D%0Aline%280%2C0%2C-3%2C-3%29%2C%0D%0Aline%280%2C0%2C4%2C-4%29%2C%0D%0Aline%28-3%2C-3%2C4%2C-4%29%2C%0D%0Aline%285%2C4%2C5%2C2%29%2C%0D%0Aline%284%2C3%2C6%2C3%29%2C%0D%0A%0D%0Alocate%284.88%2C4.52%2C%22N%22%29%2C%0D%0Alocate%286.28%2C3.2%2C%22E%22%29%2C%0D%0Alocate%284.94%2C1.76%2C%22S%22%29%2C%0D%0Alocate%283.54%2C3.22%2C%22W%22%29%2C%0D%0Alocate%28-4%2C4%2Cmatrix%281%2C3%2C%22Birds%22%2C%22eye%22%2C%22view%22%29%29%2C%0D%0Alocate%28-4%2C3.5%2Cmatrix%281%2C3%2C%22Not%22%2C%22to%22%2C%22scale%22%29%29%2C%0D%0A%0D%0Alocate%28-3.78%2C-2.8%2C%22A%22%29%2C%0D%0Alocate%28-0.16%2C0.5%2C%22C%22%29%2C%0D%0Alocate%284.26%2C-3.68%2C%22B%22%29%2C%0D%0Alocate%28-0.62%2C-3.68%2Cmatrix%281%2C2%2C%2275%22%2C%22m%22%29%29%2C%0D%0Alocate%28-1.62%2C-0.34%2C25%5Eo%29%2C%0D%0Alocate%281.06%2C-0.34%2C23%5Eo%29%0D%0A%29%0D%0A ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "A = location of the surveyor<BR>\n" );
document.write( "B = second observation location<BR>\n" );
document.write( "C = location of the tower<BR>\n" );
document.write( "Notation like W 25° S means \"aim directly west, then turn 25° toward the south\"<BR>\n" );
document.write( "The blue dashed line in the diagram above helps us set up the angles of 25° and 23°\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Refer to this page for a few other examples<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"http://academic.brooklyn.cuny.edu/geology/leveson/core/linksa/comp.html\">http://academic.brooklyn.cuny.edu/geology/leveson/core/linksa/comp.html</a>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Also, refer to this problem involving similar compass bearings notation.<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://www.algebra.com/algebra/homework/Trigonometry-basics/Trigonometry-basics.faq.question.1200155.html\">https://www.algebra.com/algebra/homework/Trigonometry-basics/Trigonometry-basics.faq.question.1200155.html</a>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Angle ACB is sandwiched between the 25 and 23 degree angles.<BR>\n" );
document.write( "Those three angles add to 180.<BR>\n" );
document.write( "25+(angleACB)+23 = 180<BR>\n" );
document.write( "angleACB + 48 = 180<BR>\n" );
document.write( "angle ACB = 180-48<BR>\n" );
document.write( "angle ACB = 132\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Let's update the drawing with that angle.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%0D%0Adrawing%28400%2C400%2C-6%2C7%2C-5%2C5%2C%0D%0Ablue%28%0D%0Aline%28-5%2C0%2C-4.75%2C0%29%2Cline%28-4.5%2C0%2C-4.25%2C0%29%2Cline%28-4%2C0%2C-3.75%2C0%29%2Cline%28-3.5%2C0%2C-3.25%2C0%29%2Cline%28-3%2C0%2C-2.75%2C0%29%2Cline%28-2.5%2C0%2C-2.25%2C0%29%2Cline%28-2%2C0%2C-1.75%2C0%29%2Cline%28-1.5%2C0%2C-1.25%2C0%29%2Cline%28-1%2C0%2C-0.75%2C0%29%2Cline%28-0.5%2C0%2C-0.25%2C0%29%2Cline%280%2C0%2C0.25%2C0%29%2Cline%280.5%2C0%2C0.75%2C0%29%2Cline%281%2C0%2C1.25%2C0%29%2Cline%281.5%2C0%2C1.75%2C0%29%2Cline%282%2C0%2C2.25%2C0%29%2Cline%282.5%2C0%2C2.75%2C0%29%2Cline%283%2C0%2C3.25%2C0%29%2Cline%283.5%2C0%2C3.75%2C0%29%2Cline%284%2C0%2C4.25%2C0%29%2Cline%284.5%2C0%2C4.75%2C0%29%0D%0A%29%2C%0D%0A%0D%0Aline%280%2C0%2C-3%2C-3%29%2C%0D%0Aline%280%2C0%2C4%2C-4%29%2C%0D%0Aline%28-3%2C-3%2C4%2C-4%29%2C%0D%0Aline%285%2C4%2C5%2C2%29%2C%0D%0Aline%284%2C3%2C6%2C3%29%2C%0D%0A%0D%0Alocate%284.88%2C4.52%2C%22N%22%29%2C%0D%0Alocate%286.28%2C3.2%2C%22E%22%29%2C%0D%0Alocate%284.94%2C1.76%2C%22S%22%29%2C%0D%0Alocate%283.54%2C3.22%2C%22W%22%29%2C%0D%0Alocate%28-4%2C4%2Cmatrix%281%2C3%2C%22Birds%22%2C%22eye%22%2C%22view%22%29%29%2C%0D%0Alocate%28-4%2C3.5%2Cmatrix%281%2C3%2C%22Not%22%2C%22to%22%2C%22scale%22%29%29%2C%0D%0A%0D%0Alocate%28-3.78%2C-2.8%2C%22A%22%29%2C%0D%0Alocate%28-0.16%2C0.5%2C%22C%22%29%2C%0D%0Alocate%28-0.46%2C-0.5%2C132%5Eo%29%2C%0D%0Alocate%284.26%2C-3.68%2C%22B%22%29%2C%0D%0Alocate%28-0.62%2C-3.68%2Cmatrix%281%2C2%2C%2275%22%2C%22m%22%29%29%2C%0D%0Alocate%28-1.62%2C-0.34%2C25%5Eo%29%2C%0D%0Alocate%281.06%2C-0.34%2C23%5Eo%29%0D%0A%29%0D%0A ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Here's a 3D look at what's going on.<BR>\n" );
document.write( "Pay close attention to the points A,B,C how they are laid out. <BR>\n" );
document.write( "Compare their relative locations to the previous diagram shown above.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%0D%0Adrawing%28400%2C400%2C-6%2C7%2C-5%2C5%2C%0D%0A%0D%0Ablue%28%0D%0Aline%282.2%2C-1.2%2C3%2C-2%29%2Cline%283%2C-2%2C2.6722%2C-1.9122%29%2Cline%283%2C-2%2C2.9122%2C-1.6722%29%2C%0D%0Alocate%28-2%2C-0.5%2C40.5%5Eo%29%2C%0D%0Alocate%281.2%2C-0.5%2C37.25%5Eo%29%2C%0D%0Aline%28-1.7%2C-1.1%2C-2.2%2C-1.6%29%2Cline%28-2.2%2C-1.6%2C-2.1451%2C-1.3951%29%2Cline%28-2.2%2C-1.6%2C-1.9951%2C-1.5451%29%0D%0A%29%2C%0D%0A%0D%0Aline%28-3%2C-2%2C4%2C-3%29%2C%0D%0Aline%284%2C-3%2C1%2C-1%29%2C%0D%0Aline%281%2C-1%2C1%2C3%29%2C%0D%0Aline%284%2C-3%2C1%2C3%29%2C%0D%0Aline%28-3%2C-2%2C1%2C3%29%2C%0D%0Aline%28-3%2C-2%2C1%2C-1%29%2C%0D%0Alocate%280%2C-2.5%2Cmatrix%281%2C2%2C%2275%22%2C%22m%22%29%29%2C%0D%0Alocate%281.26%2C0.82%2C%22x%22%29%2C%0D%0A%0D%0Alocate%28-3.32%2C-1.64%2C%22A%22%29%2C%0D%0Alocate%284.14%2C-2.68%2C%22B%22%29%2C%0D%0Alocate%280.5%2C-0.7%2C%22C%22%29%2C%0D%0Alocate%280.5%2C-1.2%2C132%5Eo%29%2C%0D%0Alocate%280.9%2C3.34%2C%22D%22%29%2C%0D%0A%0D%0Alocate%28-4%2C4%2Cmatrix%281%2C2%2C3D%2C%22view%22%29%29%2C%0D%0Alocate%28-4%2C3.5%2Cmatrix%281%2C3%2C%22Not%22%2C%22to%22%2C%22scale%22%29%29%0D%0A%0D%0A%29%0D%0A ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "triangle ABC is along the flat ground<BR>\n" );
document.write( "A = location of the surveyor<BR>\n" );
document.write( "B = second observation location<BR>\n" );
document.write( "C = base of the tower<BR>\n" );
document.write( "D = top of the tower<BR>\n" );
document.write( "x = height of the tower = length of segment CD<BR>\n" );
document.write( "angle ACB = 132 degrees<BR>\n" );
document.write( "angle CAD = 40.5 degrees<BR>\n" );
document.write( "angle CBD = 37.25 degrees\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Here is the side view of right triangle ACD<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%0D%0Adrawing%28400%2C400%2C-6%2C7%2C-5%2C5%2C%0D%0A%0D%0Ablue%28locate%28-2.2%2C-1.2%2C40.5%5Eo%29%29%2C%0D%0A%0D%0Aline%281%2C-2%2C1%2C3%29%2C%0D%0Aline%281%2C3%2C-3%2C-2%29%2C%0D%0Aline%28-3%2C-2%2C1%2C-2%29%2C%0D%0A%0D%0Alocate%281.26%2C0.82%2C%22x%22%29%2C%0D%0A%0D%0Alocate%28-3.32%2C-1.64%2C%22A%22%29%2C%0D%0Alocate%281.2%2C-1.7%2C%22C%22%29%2C%0D%0Alocate%280.9%2C3.34%2C%22D%22%29%2C%0D%0A%0D%0Aline%281%2C-1.7%2C0.7%2C-1.7%29%2C%0D%0Aline%280.7%2C-1.7%2C0.7%2C-2%29%2C%0D%0A%0D%0Alocate%28-4%2C4%2Cmatrix%281%2C2%2C%22Side%22%2C%22view%22%29%29%2C%0D%0Alocate%28-4%2C3.5%2Cmatrix%281%2C3%2C%22Not%22%2C%22to%22%2C%22scale%22%29%29%0D%0A%0D%0A%29%0D%0A ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Here is the side view of right triangle BCD<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%0D%0Adrawing%28400%2C400%2C-6%2C7%2C-5%2C5%2C%0D%0Ablue%28locate%281.8%2C-1.2%2C37.25%5Eo%29%29%2C%0D%0A%0D%0Aline%281%2C-2%2C1%2C3%29%2C%0D%0Aline%281%2C3%2C4%2C-2%29%2C%0D%0Aline%284%2C-2%2C1%2C-2%29%2C%0D%0A%0D%0Alocate%280.65%2C0.8%2C%22x%22%29%2C%0D%0Alocate%284%2C-2%2C%22B%22%29%2C%0D%0Alocate%280.65%2C-1.9%2C%22C%22%29%2C%0D%0Alocate%280.9%2C3.34%2C%22D%22%29%2C%0D%0A%0D%0Aline%281%2C-1.7%2C1.3%2C-1.7%29%2C%0D%0Aline%281.3%2C-1.7%2C1.3%2C-2%29%2C%0D%0A%0D%0Alocate%28-4%2C4%2Cmatrix%281%2C2%2C%22Side%22%2C%22view%22%29%29%2C%0D%0Alocate%28-4%2C3.5%2Cmatrix%281%2C3%2C%22Not%22%2C%22to%22%2C%22scale%22%29%29%0D%0A%0D%0A%29%0D%0A ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "At this point I get stuck. <BR>\n" );
document.write( "I don't think there's enough information to determine x.<BR>\n" );
document.write( "If we knew the length of segment AC or the length of segment BC, then we could use trigonometry to determine x.<BR>\n" );
document.write( "</font> </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );