document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1170415>Question 1170415</A>: <I><SPAN STYLE=\"word-break: break-all;\"> a man sitting on 22.5 m high storey building measures the angles of elevation of a telecom mast directly ahead of him as 43°. He measures the angle of elevation of the mast when he is 52m in the front of the building on the horizontal ground as 70°<BR>\n" );
document.write( "A calculate the height of the mast<BR>\n" );
document.write( "The distance between the building and the mast. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #851191 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><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=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=851191\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's break down this problem step-by-step.\r<BR>\n" );
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document.write( "**Diagram:**\r<BR>\n" );
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document.write( "Imagine the following:\r<BR>\n" );
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document.write( "* **B:** The base of the building.<BR>\n" );
document.write( "* **T:** The top of the building.<BR>\n" );
document.write( "* **M:** The top of the telecom mast.<BR>\n" );
document.write( "* **H:** The base of the telecom mast.<BR>\n" );
document.write( "* **P1:** The man's first position on the building.<BR>\n" );
document.write( "* **P2:** The man's second position on the ground.\r<BR>\n" );
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document.write( "**Given:**\r<BR>\n" );
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document.write( "* Building height (BT) = 22.5 m<BR>\n" );
document.write( "* Angle of elevation from P1 to M = 43°<BR>\n" );
document.write( "* Distance P2B = 52m<BR>\n" );
document.write( "* Angle of elevation from P2 to M = 70°\r<BR>\n" );
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document.write( "**Let's define:**\r<BR>\n" );
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document.write( "* Height of the mast (HM) = h<BR>\n" );
document.write( "* Distance between the building and the mast (BH) = x\r<BR>\n" );
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document.write( "**Step 1: Analyzing the first position (P1)**\r<BR>\n" );
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document.write( "* The man is on the top of the building, so TP1 is horizontal.<BR>\n" );
document.write( "* Let's draw a horizontal line from T to a point on HM, let's call it A.<BR>\n" );
document.write( "* Then, TA = BH = x, and AM = h - 22.5.<BR>\n" );
document.write( "* In triangle TAM, we have:<BR>\n" );
document.write( "    * tan(43°) = AM / TA = (h - 22.5) / x<BR>\n" );
document.write( "    * x = (h - 22.5) / tan(43°)\r<BR>\n" );
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document.write( "**Step 2: Analyzing the second position (P2)**\r<BR>\n" );
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document.write( "* In triangle HMP2, we have:<BR>\n" );
document.write( "    * tan(70°) = HM / HP2 = h / x<BR>\n" );
document.write( "    * x = h / tan(70°)\r<BR>\n" );
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document.write( "**Step 3: Equating the two expressions for x**\r<BR>\n" );
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document.write( "* Since both expressions equal x, we can set them equal to each other:<BR>\n" );
document.write( "    * (h - 22.5) / tan(43°) = h / tan(70°)<BR>\n" );
document.write( "    * (h - 22.5) * tan(70°) = h * tan(43°)<BR>\n" );
document.write( "    * h * tan(70°) - 22.5 * tan(70°) = h * tan(43°)<BR>\n" );
document.write( "    * h * tan(70°) - h * tan(43°) = 22.5 * tan(70°)<BR>\n" );
document.write( "    * h (tan(70°) - tan(43°)) = 22.5 * tan(70°)<BR>\n" );
document.write( "    * h = (22.5 * tan(70°)) / (tan(70°) - tan(43°))\r<BR>\n" );
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document.write( "**Step 4: Calculating h (height of the mast)**\r<BR>\n" );
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document.write( "* tan(70°) ≈ 2.7475<BR>\n" );
document.write( "* tan(43°) ≈ 0.9325<BR>\n" );
document.write( "* h = (22.5 * 2.7475) / (2.7475 - 0.9325)<BR>\n" );
document.write( "* h = 61.81875 / 1.815<BR>\n" );
document.write( "* h ≈ 34.06 m\r<BR>\n" );
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document.write( "**Step 5: Calculating x (distance between the building and the mast)**\r<BR>\n" );
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document.write( "* x = h / tan(70°)<BR>\n" );
document.write( "* x = 34.06 / 2.7475<BR>\n" );
document.write( "* x ≈ 12.40 m\r<BR>\n" );
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document.write( "**Therefore:**\r<BR>\n" );
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document.write( "* The height of the mast is approximately 34.06 meters.<BR>\n" );
document.write( "* The distance between the building and the mast is approximately 12.40 meters.<BR>\n" );
document.write( " </SPAN>\n" );
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