document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1150529>Question 1150529</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Building a rollercoaster\r<BR>\n" );
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document.write( "Create an equation and graph based off the following <BR>\n" );
document.write( "•must cross ground level 4 times <BR>\n" );
document.write( "•in 100 feet it must cross ground level <BR>\n" );
document.write( "•trip must last maximum of 1 minute <BR>\n" );
document.write( "•must start and end in ground level<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 #771900 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=771900\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> you're looking at a sine wave or a cosine wave.<BR>\n" );
document.write( "we'll use the sine wave.\r<BR>\n" );
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document.write( "the general formula for the sine wave is:\r<BR>\n" );
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document.write( "y = a * sin(b * (x-c)) + d\r<BR>\n" );
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document.write( "a is the amplitude<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\r<BR>\n" );
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document.write( "you can work in degrees or your can work in radians.<BR>\n" );
document.write( "since degrees are easier to understand, we'll work in degrees.\r<BR>\n" );
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document.write( "the equation i came up with is:\r<BR>\n" );
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document.write( "y = 50 * sin(24 * (x - 90/24)) + 50\r<BR>\n" );
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document.write( "the amplitude is the distance from the center line of the graph.<BR>\n" );
document.write( "when a is 50, the graph goes 50 units above the center line and 50 units below the center line.\r<BR>\n" );
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document.write( "frequency and wavelength are related by the formula:<BR>\n" );
document.write( "wavelength = 360 / frequency.<BR>\n" );
document.write( "frequency = 360 / wavelength.<BR>\n" );
document.write( "we want 4 cycles in 1 minute.<BR>\n" );
document.write( "1 minute = 60 seconds.<BR>\n" );
document.write( "4 cycles in 60 seconds means 1 full sine wave cycle in 15 seconds.<BR>\n" );
document.write( "our wavelength is therefore 15.<BR>\n" );
document.write( "the frequency becomes 360 / 15 = 24.\r<BR>\n" );
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document.write( "the formula so far is y = 50 * sin(24 * x)\r<BR>\n" );
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document.write( "the sine wave starts at the center line of the graph.<BR>\n" );
document.write( "in order for it to start at ground level, we need to shift it 90 degrees to the right.<BR>\n" );
document.write( "when the frequency is 1, the formula would be y = sin(x - 90)<BR>\n" );
document.write( "since the frequency is 24, the formula needs to be y = sin (24 * (x - 90/24))<BR>\n" );
document.write( "24 * (x - 90/24) results in 24 * x - 90 which gives us the shift of 90 degrees that we desire.\r<BR>\n" );
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document.write( "the formula so far is y = 50 * sin(24 * (x - 90/24))\r<BR>\n" );
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document.write( "ground level is asumed to be the x-axis.<BR>\n" );
document.write( "if we want the center line of the graph to be 50 units above the x-axis, we need to shift the graph up 50 units.<BR>\n" );
document.write( "that happens with d.\r<BR>\n" );
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document.write( "the formula becomes y = 50 * sin(24 * (x - 90/24)) + 50.\r<BR>\n" );
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document.write( "that's our final equation.\r<BR>\n" );
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document.write( "here's the graph.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2019/121601.jpg\" alt=\"$$$\" >\r<BR>\n" );
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document.write( "the coordinate points shown are in (x,y) format.\r<BR>\n" );
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document.write( "x is the  number of seconds.<BR>\n" );
document.write( "y is the height in feet.\r<BR>\n" );
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document.write( "when x = 0, the height is 0.<BR>\n" );
document.write( "when x = 7.5 seconds, the height is 100 feet.<BR>\n" );
document.write( "when x = 15 seconds, the height is back to 0 feet.\r<BR>\n" );
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document.write( "the roller coaster touches the ground every 15 seconds for 60 seconds.<BR>\n" );
document.write( "it therefore touches the ground 4 times in 60 seconds after it starts.\r<BR>\n" );
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document.write( "at (0,0), the height is 0 feet at 0 seconds of travel.<BR>\n" );
document.write( "at (7.5,100), the height is 100 feet at 7.5 seconds in travel.<BR>\n" );
document.write( "at (15,0), the height is 0 feet at 15 seconds in travel.<BR>\n" );
document.write( "this happens 4 times until (60,0), where the height is 0 feet at 60 seconds in travel.\r<BR>\n" );
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document.write( "every seconds, the roller coaster has gone up 100 feet and then descended back to ground level 4 times.\r<BR>\n" );
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document.write( "this same grph happens every 60 seconds.\r<BR>\n" );
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document.write( "here's the same graph going theough 3 sets of 4 cycles each.<BR>\n" );
document.write( "first set is 0 to 60.<BR>\n" );
document.write( "second set is 60 to 120.<BR>\n" );
document.write( "third set is 120 to 180.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2019/121602.jpg\" alt=\"$$$\" >\r<BR>\n" );
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document.write( "each cycle shows the fourth time the roller coaster reaches 100 feet.<BR>\n" );
document.write( "every 60 seconds, the roller coaster is back to 0 feet (ground level).\r<BR>\n" );
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document.write( "hee's a reference.\r<BR>\n" );
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