document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=112229>Question 112229</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I need help solving this question:<BR>\n" );
document.write( "From a boat on the water, the angle of elevation to the top of a cliff is 31 degrees. Another point situated 300 meters closer to the cliff has an angle of elevation of 33 degrees. What is the height of the cliff?\r<BR>\n" );
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document.write( "I was trying to draw a diagram but I'm not sure if it was right, can you please show your work so I know what I'm doing wrong? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #81891 by </B><A HREF=/tutors/aboutme.mpl?userid=bucky><B>bucky(2189)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=bucky><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=81891\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's make a sketch as follows:<BR>\n" );
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document.write( "On a piece of paper draw a horizontal line about 3 inches long. Label the left end of this<BR>\n" );
document.write( "line as point A. Label the right end of this line as point B. Next through point B draw a<BR>\n" );
document.write( "perpendicular to line AB vertically upward for a couple of inches. Label the top of this<BR>\n" );
document.write( "line as point C. Draw a line connecting point A with point C, which forms a right triangle<BR>\n" );
document.write( "ABC with line AC being the hypotenuse. Label angle BAC as 31 degrees.<BR>\n" );
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document.write( "Next, on line AB make a point about an inch to the right of point A. Label this point as<BR>\n" );
document.write( "point D. Next draw a line connecting point D with point C.  Notice that triangle BDC is another<BR>\n" );
document.write( "right triangle. Its hypotenuse is line DC. Label angle BDC as 33 degrees<BR>\n" );
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document.write( "Label the length AD as 300 meters.<BR>\n" );
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document.write( "Now understand that point A is the first position of the boat. Point C is the top of the<BR>\n" );
document.write( "cliff. Line CB represents the height of the cliff. Point D is the second position <BR>\n" );
document.write( "of the boat ... 300 meters closer to the base of the cliff. The two angles (31 degrees<BR>\n" );
document.write( "and 33 degrees) show what happens as you get closer to the base of the cliff. You have to <BR>\n" );
document.write( "have a greater angle for your line of sight to be on the top of the cliff.<BR>\n" );
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document.write( "Now let's use the tangent function. The tangent ratio is the side opposite the angle divided<BR>\n" );
document.write( "by the side adjacent to the angle.  The side opposite the 31 degree angle is side CB.<BR>\n" );
document.write( "And the side adjacent to the 31 degree angle is side AB. So we can write:<BR>\n" );
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document.write( "tan(31) = CB/AB<BR>\n" );
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document.write( "Using a calculator you can find that tan(31) = 0.600860619 <BR>\n" );
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document.write( "So the equation becomes:<BR>\n" );
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document.write( "0.600860619 = CB/AB<BR>\n" );
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document.write( "and when you multiply both sides by AB you get:<BR>\n" );
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document.write( "0.600860619AB = CB <=== call this equation #1<BR>\n" );
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document.write( "Now lets look at triangle BDC and angle BDC (33 degrees).  In this triangle, the side opposite<BR>\n" );
document.write( "angle BDC is side CB.  The side adjacent to angle BDC is side BD.  But side BD is equal<BR>\n" );
document.write( "to side AB minus 300 meters (length AD). So for the tangent of angle BDC (33 degrees) we<BR>\n" );
document.write( "get:<BR>\n" );
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document.write( "tan(33) = CB/(AB - 300)<BR>\n" );
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document.write( "Using a calculator we find that tan(33) = 0.649407593<BR>\n" );
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document.write( "Substituting this we get:<BR>\n" );
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document.write( "0.649407593 = CB/(AB - 300)<BR>\n" );
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document.write( "Multiplying both sides by (AB - 300) results in:<BR>\n" );
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document.write( "0.649407593(AB - 300) = CB<BR>\n" );
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document.write( "And multiplying out the left side results in:<BR>\n" );
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document.write( "0.649407593AB - 194.822278 = CB <==== call this equation #2<BR>\n" );
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document.write( "Notice now that equations #1 and #2 both have CB on the right side. This means that the<BR>\n" );
document.write( "left sides of these equations both equal CB and therefore, the left sides of the two equations<BR>\n" );
document.write( "must be equal. So set them equal to get:<BR>\n" );
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document.write( "0.600860619AB = 0.649407593AB - 194.822278 <BR>\n" );
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document.write( "Subtract 0.649407593AB from both sides and you get rid of the 0.649407593AB on the right<BR>\n" );
document.write( "side. The equation then becomes:<BR>\n" );
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document.write( "-0.048546974AB = -194.822278<BR>\n" );
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document.write( "Divide both sides by -0.048546974 and you get:<BR>\n" );
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document.write( "AB = -194.822278/-0.048546974 = 4013.067384 meters<BR>\n" );
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document.write( "This is the distance from the first position of the boat to the base of the cliff. Equation<BR>\n" );
document.write( "#1 applies to that position and we can substitute 4013.067384 into that equation for AB to get:<BR>\n" );
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document.write( ".600860619(4013.067384) = CB<BR>\n" );
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document.write( "Multiplying out the right side results in 2411.294153 meters = CB<BR>\n" );
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document.write( "Or the height of the cliff is about 2411 meters ... pretty tall!!<BR>\n" );
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document.write( "Hope this helps you to understand the problem and one way you can do it.<BR>\n" );
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