Design a roller coaster; a thrill ride that is just up and down with a loop. The entire ride must be comprised of calculus equations. The "thrill" of the ride must be calculated (see last parameter)

Parameters:

-The coaster must be exactly 500 feet in length along its base, leaving the ground at "zero" feet and touching back on the ground at 500 feet. It must start and end at ground level with a short horizontal section so people can get on and off the car safely-these horizontal sections are in addition to the 500 ft, so the count for the 500 ft is the point where the track leaves the ground until the track touches back down.

-The scale must be one-to-one, dont scale down roller coaster.

-The coaster will start by using a flat line before the zero mark, then a cubic or other curved function, and then build the rest by splicing together other cubic functions or other curves. The coaster must have at least one cubic function and 2 other curves (cubic or non cubic) in addition to the enclosed loop(s), for a min of 3 hills and one loop.

-The entire ride must be smooth - the function values and derivative values must match at every joint. It is easiest to splice two functions together at a min or max (why?). At least one splice must happen at a non-zero slope point.

-The coaster must include at least one loop. It is easiest to place a loop on a horizontal section of track so that it touches the track only once (why?) - anywhere else, the loop must be tangent to the curve (touching at a single point). Try using a circle or an ellipse. The loops minimum diameter must be at least 30 feet and at most, 50 feet of horizontal track will be allowed within the 500 ft length of the ride.

-There is a suppport every 25 feet, starting the measure from the point where the track leaves the ground, and the cost, in dollars, of the material needed for the support is the sum of squares of the height of EACH support, measured in feet (take the height of the support and square that quantity, then add all the squares). Supports for the loop hit the highest appropriate point. Track and cars are free but cost of the supports must be accounted for.

-The tallest hill cannot have a height greater than 150 feet. Since the first climb is the only one that has mechanical help, you must be sure that your coaster can make it over every hill. We will assume that the coaster can make it over no more then 75% of the height of the previous hill. Therefore, the tallest part of the loop cant be over 75% of the height of the previous hill. And, any peaks after the loop can only be at most 75% the final height of the loop.

-Must find the angle (in radians) at the steepest point for each descent (that is, the angle of depression) after each peak, or drop, not including the loop. This angle is defined to be the "thrill" of that decent. The "thrill" for the entire ride is sum of all the thrill decent numbners plus the number of peaks in your ride (including the loop). A loop counts as a peak, but does not have an angle. Recall the tangent trig function gives the slope of the terminal side of an angle, so if you take the inverse tangent of your slope you will get the angle measure.