document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1170834>Question 1170834</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The angle between the coastline and<BR>\n" );
document.write( "Johnson’s boat at the first lighthouse is 43°<BR>\n" );
document.write( "From the second lighthouse, the angle between the coastline and Johnson’s boat is 55° How much shorter is the direct  distance from the boat to the second lighthouse than the direct distance to the first  lighthouse? [4] <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 #851092 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1987)</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=851092\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> To solve this problem, we need to use trigonometry and the Law of Sines. Let's break it down step by step.\r<BR>\n" );
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document.write( "**1. Visualize the Problem**\r<BR>\n" );
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document.write( "Imagine a triangle formed by:\r<BR>\n" );
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document.write( "* The boat's position<BR>\n" );
document.write( "* The first lighthouse<BR>\n" );
document.write( "* The second lighthouse\r<BR>\n" );
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document.write( "Let's label the points:\r<BR>\n" );
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document.write( "* B = Boat<BR>\n" );
document.write( "* L1 = First lighthouse<BR>\n" );
document.write( "* L2 = Second lighthouse\r<BR>\n" );
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document.write( "We are given:\r<BR>\n" );
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document.write( "* ∠BL1L2 = 43°<BR>\n" );
document.write( "* ∠BL2L1 = 55°<BR>\n" );
document.write( "* The distance between the lighthouses (L1L2) is 1.5 miles.\r<BR>\n" );
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document.write( "**2. Find the Third Angle**\r<BR>\n" );
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document.write( "The sum of the angles in a triangle is 180°. Therefore:\r<BR>\n" );
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document.write( "* ∠L1BL2 = 180° - 43° - 55° = 82°\r<BR>\n" );
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document.write( "**3. Use the Law of Sines**\r<BR>\n" );
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document.write( "The Law of Sines states:\r<BR>\n" );
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document.write( "a / sin(A) = b / sin(B) = c / sin(C)\r<BR>\n" );
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document.write( "Where:\r<BR>\n" );
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document.write( "* a, b, c are the sides of the triangle<BR>\n" );
document.write( "* A, B, C are the opposite angles\r<BR>\n" );
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document.write( "Let's find the distances from the boat to each lighthouse:\r<BR>\n" );
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document.write( "* **Distance BL1 (distance to the first lighthouse):**\r<BR>\n" );
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document.write( "    * BL1 / sin(55°) = L1L2 / sin(82°)<BR>\n" );
document.write( "    * BL1 = (L1L2 * sin(55°)) / sin(82°)<BR>\n" );
document.write( "    * BL1 = (1.5 * sin(55°)) / sin(82°)<BR>\n" );
document.write( "    * BL1 ≈ (1.5 * 0.81915) / 0.99027<BR>\n" );
document.write( "    * BL1 ≈ 1.2407 miles\r<BR>\n" );
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document.write( "* **Distance BL2 (distance to the second lighthouse):**\r<BR>\n" );
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document.write( "    * BL2 / sin(43°) = L1L2 / sin(82°)<BR>\n" );
document.write( "    * BL2 = (L1L2 * sin(43°)) / sin(82°)<BR>\n" );
document.write( "    * BL2 = (1.5 * sin(43°)) / sin(82°)<BR>\n" );
document.write( "    * BL2 ≈ (1.5 * 0.68199) / 0.99027<BR>\n" );
document.write( "    * BL2 ≈ 1.0326 miles\r<BR>\n" );
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document.write( "**4. Find the Difference**\r<BR>\n" );
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document.write( "* Difference = BL1 - BL2<BR>\n" );
document.write( "* Difference ≈ 1.2407 - 1.0326<BR>\n" );
document.write( "* Difference ≈ 0.2081 miles\r<BR>\n" );
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document.write( "**5. Convert to Feet**\r<BR>\n" );
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document.write( "* 1 mile = 5280 feet<BR>\n" );
document.write( "* 0.2081 miles * 5280 feet/mile ≈ 1098.768 feet\r<BR>\n" );
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document.write( "**6. Round to the Nearest Foot**\r<BR>\n" );
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document.write( "* 1099 feet\r<BR>\n" );
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document.write( "**Answer:**\r<BR>\n" );
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document.write( "The direct distance from the boat to the second lighthouse is approximately 1099 feet shorter than the direct distance to the first lighthouse.<BR>\n" );
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