document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=478787>Question 478787</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hello\r<BR>\n" );
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document.write( "My name is Yash Shah. I am doing summer course for Algebra II and I have a following question.\r<BR>\n" );
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document.write( "If a satellite circles earth at an altitude of a = 22,300 miles, it's speed is the same as rotational speed of an earth.<BR>\n" );
document.write( "To an observer on the equator, satellite appears to be stationary.<BR>\n" );
document.write( "Using r = 4000 miles as a radius of an earth, determine the % of the equator that is within the signal range of the satellite.\r<BR>\n" );
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document.write( "Can you please assist me to resolve this problem.\r<BR>\n" );
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document.write( "Thank you.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #328066 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=328066\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> If a satellite circles earth at an altitude of a = 22,300 miles, it's speed is the same as rotational speed of an earth.<BR>\n" );
document.write( "To an observer on the equator, satellite appears to be stationary.<BR>\n" );
document.write( "Using r = 4000 miles as a radius of an earth, determine the % of the equator that is within the signal range of the satellite.<BR>\n" );
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document.write( "a picture of your situation is shown below:<BR>\n" );
document.write( "below that is the explanation.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/problems/satellite_1.jpg\" alt = \"$$$$$\"/ ><BR>\n" );
document.write( "if the satellite appears stationary to an observer on the equator, then the orbit of the satellite must be along the equator because the earth revolves around the equator.<BR>\n" );
document.write( "the picture shows the earth from a top down view so the equator looks like the circumference of the circle in the diagram.<BR>\n" );
document.write( "the satellite is at point A.<BR>\n" );
document.write( "the center of the earth is at point D.<BR>\n" );
document.write( "if you draw a straight line from the satellite to the edge of the earth, that line will be a tangent to the circle.<BR>\n" );
document.write( "the tangent to a circle is perpendicular to the radius of the circle at the point of tangency.<BR>\n" );
document.write( "also, from an external point, the tangents to both sides of the circle are equal.<BR>\n" );
document.write( "this means that:<BR>\n" );
document.write( "AB is the same length of AC.<BR>\n" );
document.write( "angle ABD and angle ACD are right angles (equal to 90 degrees).<BR>\n" );
document.write( "BD and DC are radii of the circle and are each 4000 miles long.<BR>\n" );
document.write( "the length of AD is equal to 22300 + 4000 = 26300 miles.<BR>\n" );
document.write( "that's the distance from the satellite to the center of the earth which is also the center of the circle.<BR>\n" );
document.write( "2 right triangles are formed.<BR>\n" );
document.write( "they are:<BR>\n" );
document.write( "triangle ABD and triangle ACD<BR>\n" );
document.write( "the right angles of these triangles are angle ABD and angle ACD.<BR>\n" );
document.write( "AD is the hypotenuse of each of these right triangles.<BR>\n" );
document.write( "we need to find the central angles of the circle.<BR>\n" );
document.write( "these are angle ADB and angle ADC.<BR>\n" );
document.write( "these angles are equal to each other, so if we find 1 we will have found the other.<BR>\n" );
document.write( "since these are right triangles, we can use trigonometry to find the angles.<BR>\n" );
document.write( "we'll work with triangle ACD.<BR>\n" );
document.write( "cosine (ADC) = adjacent / hypotenuse = DC/AD = 4000/26300 = .152091255<BR>\n" );
document.write( "this means that angle ADC = 81.25186267<BR>\n" );
document.write( "this means that angle ADB also equals 81.25186267 degrees because it is equal to angle ADC.<BR>\n" );
document.write( "our central angle BDC is equal to the sum of angles ADB and ADC.<BR>\n" );
document.write( "angle BDC = 162.5037253<BR>\n" );
document.write( "since the circumference of the earth has a central angle of 360 degrees, then the percentage of the circumference of the earth represented by an angle of 162.5037253 is equal to 162.5037253/360 * 100% = 45.13992371%.<BR>\n" );
document.write( "this rounds to 45.14% if you want to get to the nearest hundredth of a percent.<BR>\n" );
document.write( "the circumference of the earth is given by the equation C = 2*pi*r where r is the radius.<BR>\n" );
document.write( "this equates to 2*pi*4000 = 25,133 miles approximately.<BR>\n" );
document.write( "the section of the circumference of the earth covered by the satellite signal would be equal to 45.14% of that which would make it equal to 11,345 miles approximately. </SPAN>\n" );
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