document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1170358>Question 1170358</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hi. This is my second question for today.\r<BR>\n" );
document.write( "\n" );
document.write( "A bridge has an elliptical arch as a support. The arch has a height of 7 meters and a width at the base of 40 meters. A horizontal roadway is 2 meters above the center of the arch. How far would it be above the arch at 8 meters from the center?\r<BR>\n" );
document.write( "\n" );
document.write( "I've been solving this but can't be quite sure whether I'm correct or not. I got 0.6 m.\r<BR>\n" );
document.write( "\n" );
document.write( "Thank you so much. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #795221 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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=795221\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Draw an xy axis. Plot the point A at the origin (0,0). This will be the center of the ellipse. So (h,k) = (0,0).\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Then plot points B and C at (-20,0) and (20,0) respectively. The distance from B to C is 40 meters, which is the width of the arch.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Note the length of the semi-major axis is 20 units. The semi-major axis is horizontal, so we'll make a = 20.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Point D will go at (0,7). The semi-minor axis is vertical, so b = 7 represents the length of the semi-minor axis.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "With those h,k, a and b values, we get<BR>\n" );
document.write( "(x-h)^2/(a^2) + (y-k)^2/(b^2) = 1<BR>\n" );
document.write( "(x-0)^2/(20^2) + (y-0)^2/(7^2) = 1<BR>\n" );
document.write( "(x^2)/400 + (y^2)/49 = 1<BR>\n" );
document.write( "This represents the equation of the ellipse\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The upper roadway is represented by a horizontal line through 9 on the y axis. This is because the upper roadway is 2 meters above the highest point D. \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Here's the graph of everything discussed so far. Point E wasn't discussed earlier, but I mention it in the next section.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" width = \"50%\" src = \"https://i.imgur.com/VXYIs85.png\">\r<BR>\n" );
document.write( "\n" );
document.write( "Go back to point A. Move 8 meters either right or left. I'll move 8 meters to the right. Then you move upward to point E as shown in the diagram. Currently the location of point E is (8, p). \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "What we do is plug x = 8 into the ellipse equation we set up. Then we solve for y\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "(x^2)/400 + (y^2)/49 = 1<BR>\n" );
document.write( "(8^2)/400 + (y^2)/49 = 1<BR>\n" );
document.write( "64/400 + (y^2)/49 = 1<BR>\n" );
document.write( "4/25 + (y^2)/49 = 1<BR>\n" );
document.write( "(y^2)/49 = 1 - 4/25<BR>\n" );
document.write( "(y^2)/49 = 25/25 - 4/25<BR>\n" );
document.write( "(y^2)/49 = 21/25<BR>\n" );
document.write( "sqrt[ (y^2)/49 ] = sqrt(21/25)<BR>\n" );
document.write( "y/7 = sqrt(21)/sqrt(25)<BR>\n" );
document.write( "y/7 = sqrt(21)/5<BR>\n" );
document.write( "y = 7*sqrt(21)/5<BR>\n" );
document.write( "y = (7/5)*sqrt(21)<BR>\n" );
document.write( "y = 1.4*sqrt(21)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So we can see that E is located at ( 8, 1.4*sqrt(21) ). \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The last step will have us subtract the y coordinate of point E from y = 9, which is the height of the upper roadway. This will tell us the vertical distance from E to the upper roadway.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So we get  9 - 1.4*sqrt(21) = 2.58439402706182\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Exact Answer: 9 - 1.4*sqrt(21)  meters<BR>\n" );
document.write( "Approximate Answer: 2.58439402706182 meters<BR>\n" );
document.write( "Round the approximate value however you need.<BR>\n" );
document.write( "</font><BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );