document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1179678>Question 1179678</A>: <I><SPAN STYLE=\"word-break: break-all;\">  Two industrious Avenues students are trying to find the distance across the Pinheiros River in São<BR>\n" );
document.write( "Paulo. After marking points A and B about 86 m apart on one bank, they sight a point P, on the opposite<BR>\n" );
document.write( "bank, and measure angles PAB and PBA to be 30 and 129 degrees, respectively. This enables them to<BR>\n" );
document.write( "calculate the altitude from P to the baseline AB. To the nearest meter, how wide, straight across, is the<BR>\n" );
document.write( "Pinheiros River?<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850187 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=850187\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to solve this problem using trigonometry:\r<BR>\n" );
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document.write( "1. Visualize the Triangle:\r<BR>\n" );
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document.write( "Imagine a triangle PAB, where A and B are points on one bank of the river, and P is a point on the opposite bank.<BR>\n" );
document.write( "We know AB = 86 meters.<BR>\n" );
document.write( "We know angle PAB = 30 degrees.<BR>\n" );
document.write( "We know angle PBA = 129 degrees.<BR>\n" );
document.write( "2. Calculate Angle APB:\r<BR>\n" );
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document.write( "The sum of angles in a triangle is 180 degrees.<BR>\n" );
document.write( "Angle APB = 180 - (30 + 129) = 180 - 159 = 21 degrees.<BR>\n" );
document.write( "3. Use the Law of Sines to Find AP:\r<BR>\n" );
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document.write( "The Law of Sines states: a/sin(A) = b/sin(B) = c/sin(C)<BR>\n" );
document.write( "In our case: AP/sin(PBA) = AB/sin(APB)<BR>\n" );
document.write( "AP/sin(129) = 86/sin(21)<BR>\n" );
document.write( "AP = (86 * sin(129)) / sin(21)<BR>\n" );
document.write( "AP ≈ (86 * 0.7771) / 0.3584<BR>\n" );
document.write( "AP ≈ 234.8 meters<BR>\n" );
document.write( "4. Calculate the Altitude (Width of the River):\r<BR>\n" );
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document.write( "Let h be the altitude from P to AB.<BR>\n" );
document.write( "In triangle AP, we have: sin(PAB) = h/AP<BR>\n" );
document.write( "sin(30) = h/234.8<BR>\n" );
document.write( "h = 234.8 * sin(30)<BR>\n" );
document.write( "h = 234.8 * 0.5<BR>\n" );
document.write( "h = 117.4 meters<BR>\n" );
document.write( "5. Round to the Nearest Meter:\r<BR>\n" );
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document.write( "The width of the river is approximately 117 meters.<BR>\n" );
document.write( "Final Answer:\r<BR>\n" );
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document.write( "The width of the Pinheiros River is approximately 117 meters. </SPAN>\n" );
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