document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1162369>Question 1162369</A>: <I><SPAN STYLE=\"word-break: break-all;\">  Suppose you wanted to model a ferris wheel using a sine function that took 60<BR>\n" );
document.write( "seconds to complete one revolution. The ferris wheel must start 0.5 m above ground.\r<BR>\n" );
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document.write( "Provide an equation of such a sine function that will ensure that the ferris wheel’s minimum height of the ground is 0.5 m.<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #786156 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=786156\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> the general form of the sine function is y = a * sin(b * (x - c)) + d<BR>\n" );
document.write( "a is the amplitube<BR>\n" );
document.write( "b is the frequency <BR>\n" );
document.write( "c is the horizontal shift<BR>\n" );
document.write( "d is the vertical shift.<BR>\n" );
document.write( "formula for frequency is f = 360 / p<BR>\n" );
document.write( "f is the frequency<BR>\n" );
document.write( "p is the period<BR>\n" );
document.write( "if you want one full sine wave to be 60 seconds, then let 60 degrees represent 60 seconds.<BR>\n" );
document.write( "your period is 60 seconds.<BR>\n" );
document.write( "f = 360 / 60 = 6<BR>\n" );
document.write( "if your frequency is 6, then you will get 6 full sine waves in 360 degrees.<BR>\n" );
document.write( "that would make each full sine wave equal to 60 degrees.<BR>\n" );
document.write( "with b = 6, your sine wave function becomes y = a * sin(6 * (x - c)) + d<BR>\n" );
document.write( "when a = 1, the sine wave is plus or minus 1 unit from the center line of the sine wave.<BR>\n" );
document.write( "that makes the top 1 unit above the center line and the bottom 1 unit below the center line.<BR>\n" );
document.write( "the center line of the sine wave is the x-axis without any adjustment.<BR>\n" );
document.write( "if you let the x-axis represent the ground and you want the bottom of the sine wave to be .5 units above the ground, then d must be equal to 1.5<BR>\n" );
document.write( "that raises the center line of the sine wave to y = 1.5 which means the bottom of the sine wave will be at y = .5 which is .5 units above the ground.<BR>\n" );
document.write( "set d to 1.5 and the formula becomes y = a * sin(6 * (x - c)) + 1.5<BR>\n" );
document.write( "let's take a look at whaat we have so far.<BR>\n" );
document.write( "since c is normally 0, we set it to 0.<BR>\n" );
document.write( "since a is normall 1, we set it to 1.<BR>\n" );
document.write( "the function becomes y = 1 * sin(6 * (x - 0)) + 1.5<BR>\n" );
document.write( "this is what it looks like on the graph.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2020/071501.jpg\" >\r<BR>\n" );
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document.write( "the sine wave starts at the center line which is 1.5 units above the x-axis which represents the ground.<BR>\n" );
document.write( "we want the sine wave to start at .5 units above the ground.<BR>\n" );
document.write( "from the graph we can see that if we shift the sine wave to the right 15 degrees, then it will start .5 units above the ground.<BR>\n" );
document.write( "we need a horizontal shift to the right of 15 degrees.<BR>\n" );
document.write( "shift to the right means c must be positive.<BR>\n" );
document.write( "therefore we make c = 15 and the formula becomes y = 1 * sin(6 * (x - 15)) + 1.5<BR>\n" );
document.write( "this is what that looks like on the graph.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2020/071502.jpg\" >\r<BR>\n" );
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document.write( "the final diagram is what you want.<BR>\n" );
document.write( "one full cycle is 60 degrees which represents 60 seconds.<BR>\n" );
document.write( "the cycle starts .5 meters above the ground.<BR>\n" );
document.write( "the a point on the ferris wheel has a minimum height of .5 meters above the ground and a maximum height of 2.5 meters above the ground.<BR>\n" );
document.write( "the amplitude of the sine plus or minus 1 unit from the center line which is at 1.5 meters above the ground.<BR>\n" );
document.write( "the point on the ferris wheel that was .5 meters from the ground when it starts will be back to .5 meters above the ground every 60 seconds.\r<BR>\n" );
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document.write( "here's a view of what the sine wave looks like after it went through three 60 second cycles.\r<BR>\n" );
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document.write( "your final equation was y = 1 * sin(6 * (x-15)) + 1.5\r<BR>\n" );
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document.write( "since a = 1 is default, the equation can be written as y = sin(6 * (x-15)) + 1.5\r<BR>\n" );
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document.write( "here's a reference on the equation to graph a sine wave.\r<BR>\n" );
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document.write( "<a  rel=nofollow HREF= \"https://mathbitsnotebook.com/Algebra2/TrigGraphs/TGsinusoidal.html\" target = \"_blank\">https://mathbitsnotebook.com/Algebra2/TrigGraphs/TGsinusoidal.html</a> </SPAN>\n" );
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