document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=468828>Question 468828</A>: <I><SPAN STYLE=\"word-break: break-all;\"> On a 3% railroad grade, at what angle are the rails inclined to the horizontal, and how far does one rise in traveling upward 5000 feet measured along the rails? (The track rise 3 feet for each 100 feet of horizontal distance gained.) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #321662 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=321662\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> it looks like the 3% grade is referring to the slope of line made by the tracks.<BR>\n" );
document.write( "the railroad people describe this as the rise to run ratio.<BR>\n" );
document.write( "the grade is expressed as a percent which is the ratio times 100%.<BR>\n" );
document.write( "you are told that the track line rises 3 feet for every 100 feet of horizontal travel.<BR>\n" );
document.write( "this makes the grade equal to 3/100 = .03 * 100% = 3%.<BR>\n" );
document.write( "you can use this information to find the angle of elevation.<BR>\n" );
document.write( "draw a right triangle and label it ABC.  A is on the left, B is on the top right, C is on the right at the same level as A.<BR>\n" );
document.write( "The angle of elevation is angle A.<BR>\n" );
document.write( "The vertical distance traveled is BC.<BR>\n" );
document.write( "the horizontal distance traveled is AC.<BR>\n" );
document.write( "the distance along the track is AB.<BR>\n" );
document.write( "AB is the hypotenuse of this triangle.<BR>\n" );
document.write( "AC is the side adjacent to angle A.<BR>\n" );
document.write( "BC is the side opposite to angle A.<BR>\n" );
document.write( "you use the tangent formula to find angle A.<BR>\n" );
document.write( "tangent (A) = opposite / adjacent = 3/100 = .03<BR>\n" );
document.write( "A is the angle whose tangent is .03<BR>\n" );
document.write( "A is equal to 1.718358002 degrees.<BR>\n" );
document.write( "That's your angle of elevation.<BR>\n" );
document.write( "you can now use that angle to find the vertical distance after the train has traveled 5000 feet along the track bed.<BR>\n" );
document.write( "the track bed is the hypotenuse of the triangle.<BR>\n" );
document.write( "the horizontal distance traveled is the adjacent side to angle A.<BR>\n" );
document.write( "the vertical distance traveled is the opposite side to angle A.<BR>\n" );
document.write( "you will use the sine formula to find the vertical distance traveled.<BR>\n" );
document.write( "sine (A) = opposite / hypotenuse = x / 5000<BR>\n" );
document.write( "multiply both sides of this equation by 5000 to get:<BR>\n" );
document.write( "5000 * sine (1.718358002) = x which gets you:<BR>\n" );
document.write( "x = 149.9325455 feet<BR>\n" );
document.write( "you will use the cosine formula to find the horizontal distance traveled.<BR>\n" );
document.write( "cosine (A) = adjacent / hypotenuse = y / 5000<BR>\n" );
document.write( "multiply both sides of this equation by 5000 to get:<BR>\n" );
document.write( "5000 * cosine (1.718358002) = y which gets you:<BR>\n" );
document.write( "y = 4997.751518 feet.<BR>\n" );
document.write( "your figures are now:<BR>\n" );
document.write( "angle A = 1.718358002 degrees<BR>\n" );
document.write( "distance along track bed traveled = hypotenuse = 5000 feet.<BR>\n" );
document.write( "vertical distance traveled = elevation = opposite side = 149.<BR>\n" );
document.write( "horizontal distance traveled = adjacent side = 4997.751518 feet.<BR>\n" );
document.write( "as a test, take the vertical distance traveled and divide it by 100 to get 49.97751518 feet.<BR>\n" );
document.write( "we then multiply that by 3 to get 149.9325455 feet.<BR>\n" );
document.write( "this is exactly the vertical distance we calculated earlier using the sine formula, only this time we used the 3% grade formula of 3 / 100 to get it.<BR>\n" );
document.write( "answer check out so we're good.<BR>\n" );
document.write( "answers to your question are:<BR>\n" );
document.write( "angle = 1.718358002 degrees<BR>\n" );
document.write( "vertical distance for 5000 feet of travel along the track bed = 149.9325455 feet.<BR>\n" );
document.write( "round to whatever number you need.<BR>\n" );
document.write( "a picture of what we just did is shown below:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/problems/railroad_1.jpg\" alt = \"***** picture not found *****\"/ >\r<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );