Question

Suppose you are asked to design the first ascent and drop for a new roller coaster. By studying photographs of your favorite coasters, you decide to make the slope of the ascent 0.8 and the slope of the drop -1.6. You decide to connect these two straight stretches y= L1(x) and y= L2(x) with part of a parabola y = f(x) = ax^2+ bx + c, where x and f(x) are measured in feet. For the track to be smooth there can’t be abrupt changes in direction, so you want the linear segments L1 and L2 to be tangent to the parabola at the transition points P and Q. (See the figure.) To simplify the equations, you decide to place the origin at P.

(a) Suppose the horizontal distance between P and Q is 100 ft. Write equations in a, b, and c that will ensure that the track is smooth at the transition points.

(b) Solve the equations in part (a) for a, b, and c to find a formula for f(x)

(c) Plot L1, f, and L2 to verify graphically that the transitions are smooth.

d) Find the difference in elevation between P and Q.

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Answer 1

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