document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1186123>Question 1186123</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The angular elevation of a tower CD at a place A due south of it is 60°; and at a place B due west of A, the elevation is 30°. If AB = 3 km, what is the height of the tower? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849403 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=849403\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Let points A,B,C be placed on the ground. <BR>\n" );
document.write( "A and B are the observation points.<BR>\n" );
document.write( "C is the base of the tower.<BR>\n" );
document.write( "D is the top of the tower.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%0D%0Adrawing%28400%2C400%2C-3%2C7%2C-5%2C5%2C%0D%0Acircle%281%2C-2%2C0.06%29%2Ccircle%281%2C-2%2C0.08%29%2Ccircle%281%2C-2%2C0.1%29%2Ccircle%281%2C-2%2C0.12%29%2Ccircle%28-2%2C-2%2C0.06%29%2Ccircle%28-2%2C-2%2C0.08%29%2Ccircle%28-2%2C-2%2C0.1%29%2Ccircle%28-2%2C-2%2C0.12%29%2Ccircle%281%2C2%2C0.06%29%2Ccircle%281%2C2%2C0.08%29%2Ccircle%281%2C2%2C0.1%29%2Ccircle%281%2C2%2C0.12%29%2Ccircle%283%2C-2%2C0.06%29%2Ccircle%283%2C-2%2C0.08%29%2Ccircle%283%2C-2%2C0.1%29%2Ccircle%283%2C-2%2C0.12%29%2Ccircle%285%2C-2%2C0.06%29%2Ccircle%285%2C-2%2C0.08%29%2Ccircle%285%2C-2%2C0.1%29%2Ccircle%285%2C-2%2C0.12%29%2Ccircle%286.04%2C-0.96%2C0.06%29%2Ccircle%286.04%2C-0.96%2C0.08%29%2Ccircle%286.04%2C-0.96%2C0.1%29%2Ccircle%286.04%2C-0.96%2C0.12%29%2Ccircle%286%2C2%2C0.06%29%2Ccircle%286%2C2%2C0.08%29%2Ccircle%286%2C2%2C0.1%29%2Ccircle%286%2C2%2C0.12%29%2C%0D%0A%0D%0Aline%281%2C-2%2C1%2C2%29%2Cline%281%2C2%2C-2%2C-2%29%2Cline%28-2%2C-2%2C1%2C-2%29%2Cline%286%2C-1%2C3%2C-2%29%2Cline%286%2C-1%2C5%2C-2%29%2Cline%286%2C-1%2C6%2C2%29%2Cline%283%2C-2%2C5%2C-2%29%2Cline%283%2C-2%2C6%2C2%29%2Cline%285%2C-2%2C6%2C2%29%2C%0D%0A%0D%0Alocate%281.1%2C-2.1%2C%22A%22%29%2Clocate%28-1.9%2C-2.1%2C%22B%22%29%2Clocate%281.1%2C1.9%2C%22C%22%29%2Clocate%283.1%2C-2.1%2C%22B%22%29%2Clocate%285.1%2C-2.1%2C%22A%22%29%2Clocate%286.14%2C-1.06%2C%22C%22%29%2Clocate%286.1%2C1.9%2C%22D%22%29%2C%0D%0Alocate%281.2%2C0.08%2Cx%29%2Clocate%28-0.84%2C-2.26%2C3%29%2Clocate%28-0.88%2C0.18%2Cy%29%2Clocate%284-0.5%2C-1.22%2B0.15%2C30%5Eo%29%2Clocate%286.26%2C1.2%2Ch%29%2Clocate%285.72%2C-1.5%2C60%5Eo%29%2C%0D%0A%0D%0Alocate%283%2C-3%2Cmatrix%281%2C3%2C%22angleCAD%22%2C%22=%22%2C60%5Eo%29%29%2C%0D%0Alocate%283%2C-3.5%2Cmatrix%281%2C3%2C%22angleCBD%22%2C%22=%22%2C30%5Eo%29%29%2C%0D%0A%0D%0Alocate%28-2%2C3%2Cmatrix%281%2C3%2C%22Birds%22%2C%22Eye%22%2C%22View%22%29%29%2C%0D%0Alocate%284%2C3%2Cmatrix%281%2C2%2C3D%2C%22View%22%29%29%2C%0D%0Alocate%28-3%2C-4.5%2Cmatrix%281%2C5%2C%22Diagrams%22%2C%22are%22%2C%22not%22%2C%22to%22%2C%22scale%22%29%29%0D%0A%29%0D%0A ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( "x = distance from A to C<BR>\n" );
document.write( "y = distance from B to C<BR>\n" );
document.write( "h = height of the tower = distance from C to D<BR>\n" );
document.write( "Each distance is in kilometers and each is positive.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Focus on right triangle CAD.<BR>\n" );
document.write( "tan(angle) = opposite/adjacent<BR>\n" );
document.write( "tan(angle CAD) = CD/CA<BR>\n" );
document.write( "tan(60) = h/x<BR>\n" );
document.write( "h = x*tan(60)<BR>\n" );
document.write( "h = x*sqrt(3)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Now focus on right triangle CBD.<BR>\n" );
document.write( "tan(angle) = opposite/adjacent<BR>\n" );
document.write( "tan(angle CBD) = CD/CB<BR>\n" );
document.write( "tan(30) = h/y<BR>\n" );
document.write( "h = y*tan(30)<BR>\n" );
document.write( "x*sqrt(3) = y*tan(30) ........... plugged in h = x*sqrt(3)<BR>\n" );
document.write( "x*sqrt(3) = y*(1/sqrt(3))<BR>\n" );
document.write( "y = x*sqrt(3)*sqrt(3)<BR>\n" );
document.write( "y = 3x\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Focus on right triangle ABC that is entirely on the ground.<BR>\n" );
document.write( "The horizontal leg (that runs east/west) is AB = 3 km.<BR>\n" );
document.write( "The leg running north/south is AC = x and the hypotenuse is BC = y = 3x\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Use the Pythagorean Theorem for right triangle ABC.<BR>\n" );
document.write( "(AB)^2 + (AC)^2 = (BC)^2<BR>\n" );
document.write( "(3)^2 + (x)^2 = (3x)^2<BR>\n" );
document.write( "9 + x^2 = 9x^2<BR>\n" );
document.write( "9x^2-x^2 = 9<BR>\n" );
document.write( "8x^2 = 9<BR>\n" );
document.write( "x^2 = 9/8<BR>\n" );
document.write( "x = sqrt(9/8)<BR>\n" );
document.write( "Since x > 0 we ignore the minus of the plus/minus.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Then,<BR>\n" );
document.write( "h = x*sqrt(3)<BR>\n" );
document.write( "h = sqrt(9/8)*sqrt(3)<BR>\n" );
document.write( "h = sqrt((9/8)*3)<BR>\n" );
document.write( "h = sqrt(27/8)<BR>\n" );
document.write( "h = sqrt(27)/sqrt(8)<BR>\n" );
document.write( "h = (3*sqrt(3))/(2*sqrt(2))<BR>\n" );
document.write( "h = (3*sqrt(3)*sqrt(2))/(2*sqrt(2)*sqrt(2)) .... rationalizing denominator<BR>\n" );
document.write( "h = (3*sqrt(3*2))/(2*2)<BR>\n" );
document.write( "h = <font color=red>(3/4)*sqrt(6) km</font> is the exact answer<BR>\n" );
document.write( "h = <font color=red>1.837117 km approximately</font>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The approximate answer varies depending how you round. \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "------------------------------------------------------------------------------\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Another way to solve:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Focus on triangle CAD<BR>\n" );
document.write( "tan(60) = h/x<BR>\n" );
document.write( "x = h/tan(60)<BR>\n" );
document.write( "x = h/sqrt(3)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Focus on triangle CBD<BR>\n" );
document.write( "tan(30) = h/y<BR>\n" );
document.write( "y = h/tan(30)<BR>\n" );
document.write( "y = h*sqrt(3)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The last triangle to focus on is ABC.<BR>\n" );
document.write( "It has legs AB = 3 and AC = x. The hypotenuse is BC = y.<BR>\n" );
document.write( "Use the Pythagorean Theorem<BR>\n" );
document.write( "(AB)^2 + (AC)^2 = (BC)^2<BR>\n" );
document.write( "3^2 + x^2 = y^2<BR>\n" );
document.write( "3^2 + (h/sqrt(3))^2 = (h*sqrt(3))^2<BR>\n" );
document.write( "9 + (h^2)/3 = h^2*3<BR>\n" );
document.write( "27 + h^2 = 9h^2<BR>\n" );
document.write( "9h^2-h^2 = 27<BR>\n" );
document.write( "8h^2 = 27<BR>\n" );
document.write( "h^2 = 27/8<BR>\n" );
document.write( "h = sqrt(27/8)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The steps from here are the same as the previous section.<BR>\n" );
document.write( "You should arrive at<BR>\n" );
document.write( "h = <font color=red>(3/4)*sqrt(6) km</font> = <font color=red>1.837117 km approximately</font> \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "As of 2025, the world's tallest building is the Burj Khalifa at around 828 meters tall. <BR>\n" );
document.write( "The answer we got is over 1000 meters taller, so tower CD is definitely breaking records.<BR>\n" );
document.write( "I don't know how realistic it is for a tower to be this tall. <BR>\n" );
document.write( "</font> </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );