2 replies to this topic

[itsme1]

This project is due Monday, November 1st 2010. I have the first question, but I just need help getting started on the second question. I am in Calculus 1. Senior year in high school (grade 12).

Here is the question:

Quote

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=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 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 L₁, f, and L₂ 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 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
  
  h(x)=px³ + qx² +rx +n

  (a) Write a system of equations in 11 unknowns that ensure that the functions and their first two derivatives agree at the transition points.
  (b) **We don't have to do part (b)**
  ( c) Plot L₁, g, q, h, and L₂, 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:

Quote

Equation for the parabola: f(x)= -.012x²+.8x
Equation for tangent line L₁= -.8x
Equation for tangent line L₂= -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.

[Rashy]

The key things to note:

At point x=0, values of L1 and g(x) are equal, as well as the derivatives and second derivatives.
At point x=10, g(10) = q(10); and g'(10) = q'(10); second derivatives as well.
At point x = 90, q(90) = h(90); q'(90) = h'(90) and same holds true for second derivatives.
At x=100, h(100) = L2(100), and the first/second derivatives agree. In the case of both lines, the second derivative is 0, which will help you determine g and h, along with the other equations that equate these points.

You should have 12 equations, one of which isn't needed, or looks identical to one of the other 11.

It's likely the parabola you solved for in question 1 still holds from 10<x<90, but you can go back to square 1 and use the given q(x)

[itsme1]

El Ragingski, on 31 October 2010 - 08:13 AM, said:

Ahh tough luck lol, because this question could be easily solved as a Non-Uniform Circular Motion (Physics)

And there are physics questions that could easily be solved with Calculus. Such is life.

Thanks for the help rashy; it was, well, helpful.

Edited by itsme1, 01 November 2010 - 10:40 PM.