Here is the question:

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 stretches y=L_{1}(x) and y=L_{2}(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 L_{1}and L_{2}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.

- 1.(a) Suppose the horizontal distance between P and Q is 100ft. 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 L_{1}, f, and L_{2}to verify graphically that the transitions are smooth.

(d) Find the Difference in elevation between P and Q.

------------------------------------------------------------------------------------------------ 2. The solution in Problem 1 might look smooth, but it might not feel smooth because the piecewise defined function [consisting of L
_{1}(x) for x<0, f(x) for 0<x<100, and L_{2}(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)=ax^{2}+bx+c only on the interval 10<x<90 and connecting it to the linear functions by means of two cubic functions:

g(x)=kx

(a) Write a system of equations in 11 unknowns that ensure that the functions and their first two derivatives agree at the transition points.^{3}+ lx^{2}+mx +n

h(x)=px^{3}+ qx^{2}+rx +n

(b) **We don't have to do part (b)**

( c) Plot L_{1}, g, q, h, and L_{2}, and compare with the plot in problem 1©.

I have all of part one. Here are the parts that will be needed to solve for part 2:

Equation for the parabola: f(x)= -.012x

^{2}+.8x

Equation for tangent line L_{1}= -.8x

Equation for tangent line L_{2}= -1.6x+120

P=(0,0)

Q=(100, -40)

I think that is all that will be needed. I really only need help getting started on part 2 (a). I do not want the problem done for me. Thanks. :)

Here is the image given in the book:

**Edited by itsme1, 30 October 2010 - 05:16 PM.**