document.write( "Question 823614: So my teacher has asked us to create an equation for a \"roller coaster\" that has the following requirements:
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document.write( "-Your coaster ride must have at least 3 relative maxima and/or minima
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document.write( "-the ride length must be at least 4 minutes
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document.write( "-the coaster ride starts at 250 feet
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document.write( "The ride dives below the ground into a tunnel at least once\r
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document.write( "Any help would be greatly appreciated. I am so lost and it is due tomorrow. Thanks! \n" );
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Algebra.Com's Answer #495837 by KMST(5328)![]() ![]() You can put this solution on YOUR website! A ride longer than 3 minutes would be unusual, but this is not real life roller coaster design. \n" ); document.write( "The simplest answer would be a polynomial function \n" ); document.write( "A polynomial of higher degree may work nicely too. \n" ); document.write( "A graphing calculator, or graphing software would allow testing different roller coaster designs, until a nice one is found. \n" ); document.write( "The domain of the function must be specified as \n" ); document.write( "Ground level would be \n" ); document.write( "Going before ground level once, the coaster should pass by ground level twice: on the way down and again on the way up. That means two real zeros for the polynomial. \n" ); document.write( " \n" ); document.write( "WITHOUT A GRAPHING CALCULATOR: \n" ); document.write( "I would try to make a degree 4 polynomial by multiplying two quadratic polynomials. \n" ); document.write( " \n" ); document.write( "It downhill fast at \n" ); document.write( "A factor like that may be a good start for the ride. \n" ); document.write( "We could also use any other quadratic equation. They all have one maximum (or one minimum, if the leading coefficient is negative. \n" ); document.write( "The function \n" ); document.write( " \n" ); document.write( " \n" ); document.write( "The product of the \n" ); document.write( " \n" ); document.write( "There are many 4th degree polynomials with a positive leading coefficient that would work. You need to make sure that they have two zeros in the (0,end of domain) interval (end of domain>=4), that there are 2 maxima in that interval, and that there is a negative minimum. \n" ); document.write( "It would also be nice if \n" ); document.write( "ON second thought, it would also be more realistic if the roller coaster never got as high as 250 feet again, because what moves is only gravity. \n" ); document.write( " \n" ); document.write( " \n" ); document.write( "would work, because the second minimum happens at \n" ); document.write( " \n" ); document.write( " \n" ); document.write( "That ride does not look like much fun, and with a degree 4 polynomial, the riders would be getting off at a point where the track is uphill. \n" ); document.write( " \n" ); document.write( "WITH A GRAPHING CALCULATOR OR SOFTWARE: \n" ); document.write( "With some help for graphing, we could have a better design. \n" ); document.write( "Maybe we should aim for a degree 5 polynomial and let the riders out at the next maximum. \n" ); document.write( "Something based on \n" ); document.write( " \n" ); document.write( "Multiplied times \n" ); document.write( " \n" ); document.write( "However, it does not dip too far below zero, and it does not rise too high during the ride. It would not be fun. Besides, the coaster would go below ground twice, and maybe the problem meant to say \"just once\". \n" ); document.write( " \n" ); document.write( "Because the first 2 zeros are close together, the first dip does not go too low. \n" ); document.write( "Adding 2, that first dip does not go below zero, and the value for \n" ); document.write( "Next, multiplying times \n" ); document.write( " \n" ); document.write( " \n" ); document.write( "with \n" ); document.write( "That second maximum could be the end of the ride. \n" ); document.write( " \n" ); document.write( "However, the lowest minimum would have \n" ); document.write( " \n" ); document.write( "An improved version , \n" ); document.write( "Adding 12, the first dip does not go below zero, and the value for \n" ); document.write( "Next, multiplying times \n" ); document.write( " \n" ); document.write( " \n" ); document.write( "with \n" ); document.write( "That second maximum could be the end of the ride. \n" ); document.write( " |