Building a better roller coaster
Calculus.. Building a better roller coaster?
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 streches y = L1(x) and y = L2(x) with part of a parabola y = f(x) = ax2 + 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 (from L1 to the parabola) and Q (from the parabola to L2). To simplify the equations you decide to place the origin at P.
1-a) Suppose the horizontal distance between P and Q is 100 feet. Write equations in a, b, and c that will ensure that the track is smooth at the transition points.
1-b) Solve the equations in part (a) for a, b, and c to find a formula for f(x).
1-c) Find the difference in elevation between P and Q.
The solution in Problem 1 might look smooth, but it might not feel smooth because the piecewise defined function (consisting of L1(x) for x < 0, f(x) for 0 < x < 100, and L2(x) for x > 100) doesn't have a continuous second derivative. So you decide to improve the design by using a quadratic function q(x) = ax2 + bx + c only on the interval 10 < x < 90 and connecting it to the linear functions by means of two cubic functions:
g(x) = kx3 + lx2 + mx + n for 0 < x < 10
h(x) = px3 + qx2 + rx + s for 90 < x < 100
2-a) Write a system of equations in 11 unknowns that ensure that the functions and their first two derivatives agree at the transition points.
2-b) Solve the equations in part (a) with a computer algebra system to find formulas for q(x), g(x), and h(x).
Particularly help with PART 2 Please....