document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=237297>Question 237297</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A ferris wheel is 40 meters in diameter and boarded at ground level. The wheel makes one full rotation every 7 minutes, and at time t=0  you are at the 3 o'clock position and ascending. Let f(t) denote your height (in meters) above ground at t  minutes. Find a formula for f(t).  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #174521 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=174521\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> ferris wheel is 40 meters in diameter.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "that means the radius is 20 meters.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the circumference of a circle is given by the equation 2*pi*r\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "it takes 7 minutes to make a complete circle.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "each minute will therefore take you 360/7 degrees around the circle.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "that's equal to 51.42857143 degrees\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "when t = 0, you are at 3 o'clock position which is exactly 20 feet above the ground.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "when t = 7/4, you are exactly at the top of the ferris wheel which make you 40 feet above the ground.  7/4 * 360/7 = 360/4 = 90 degrees.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "when t = 14/4, you are at the 9 o'clock position which is exactly 20 feet above the ground on the opposite side that you were when t = 0.  when t = 14/4, you are at exactly 14/4 * 360/7 = 180 degrees.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "when t = 21/4, you are at ground level which winds up being 270 degrees.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "when t = 28/4, you are back at the 3 o'clock position because you just made one full turn.  28/4 equals 7 minutes because 7*4 = 28.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "bottom line is the ferris wheel is a simulation of your x and y coordinates where the x-axis is at the 9 o'clock and 3 o'clock position, and the y-axis is at the 12 o'clock and 6 o'clock position.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "you can simulate where you will be by taking the sin of the angles that you will be at in a specified period of time.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the angle is specified as 360/7 * t which is in minutes.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the elevation is 20 plus the radius of the ferris wheel * the sin of the angle.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the formula would be:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "h = f(t) = 20 * (1 + sin(x)) where:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "x = 360/7 * t<BR>\n" );
document.write( "t = minutes<BR>\n" );
document.write( "h = the height you will be at.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the formula says that h is a function of t and is equal to 20 * (1+sin(x)).\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "let's see how this works.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "when t = 0 which is 0 minutes, then:<BR>\n" );
document.write( "x = 360/7 * t = 360/7 * 0 = 0 degrees<BR>\n" );
document.write( "h = f(t) = 20 * (1 + sin(x)) becomes:<BR>\n" );
document.write( "h = f(0) = 20 * (1 + sin(0))<BR>\n" );
document.write( "since sin(0) = 0, this becomes:<BR>\n" );
document.write( "h = f(t) = 20 * (1 + 0) = 20 * 1 = 20<BR>\n" );
document.write( "you are 20 feet above the ground and you are at the 3 o'clock position of the ferris wheel which is half the way to the top.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "when t = 7/4 which is 1.75 minutes, then:<BR>\n" );
document.write( "x = 360/7 * t = 360/7 * 7/4 = 360/4 = 90 degrees<BR>\n" );
document.write( "h = f(t) = 20 * (1 + sin(t)) becomes<BR>\n" );
document.write( "h = f(7/4) = 20 * (1 + sin(90))<BR>\n" );
document.write( "since sin(90) = 1, this becomes:<BR>\n" );
document.write( "h = f(7/4) = 20 * (1 + 1) = 20 * 2 = 40<BR>\n" );
document.write( "you are 40 feet above the ground and you are at the 12 o'clock position of the ferris wheel which is all the way to the top.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "when t = 14/4 which is 3.5 minutes, then:<BR>\n" );
document.write( "x = 360/7 * t = 360/7 * 14/4 = 360*2/4 = 720/4 = 180 degrees<BR>\n" );
document.write( "h = f(t) = 20 * (1 + sin(t)) becomes<BR>\n" );
document.write( "h = f(14/4) = 20 * (1 + sin(180))<BR>\n" );
document.write( "since sin(180) = 0, this becomes:<BR>\n" );
document.write( "h = f(14/4) = 20 * (1 + 0) = 20 * 1 = 20<BR>\n" );
document.write( "you are 20 feet above the ground and you are at the 9 o'clock position of the ferris wheel which is half the way to the top.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "when t = 21/4 which is 5.25 minutes, then:<BR>\n" );
document.write( "x = 360/7 * t = 360/7 * 21/4 = 360*3/4 = 1080/4 = 270 degrees<BR>\n" );
document.write( "h = f(t) = 20 * (1 + sin(t)) becomes<BR>\n" );
document.write( "h = f(21/4) = 20 * (1 + sin(270))<BR>\n" );
document.write( "since sin(270) = -1, this becomes:<BR>\n" );
document.write( "h = f(14/4) = 20 * (1 - 1) = 20 * 0 = 0<BR>\n" );
document.write( "you are 0 feet above the ground and you are at the 6 o'clock position of the ferris wheel which is at the bottom at ground level.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "when t = 6 minutes, then:<BR>\n" );
document.write( "x = 360/7 * 6 = 2160/7 = 308.5714286 degrees<BR>\n" );
document.write( "h = f(t) = 20 * (1 + sin(t)) becomes<BR>\n" );
document.write( "h = f(6) = 20 * (1 + sin(308.5714286))<BR>\n" );
document.write( "since sin(308.5714286) = -.781831482, this becomes:<BR>\n" );
document.write( "h = f(6) = 20 * (1 - .781831482) = 20 * .218168518 = 4.36337351<BR>\n" );
document.write( "you are 4.36337351 feet above the ground and you are somewhere between the 4 o'clock and 5 o'clock position on the ferris wheel, much closer to the 5 o'clock position than the 4 o'clock position.   This would be roughly 4:43 or a quarter to 5.  we are talking about the hour hand position.    \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "looks like the formula works.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "if you need a picture, let me know and I'll send you one.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );