NOT Looking for an Answer - Length of Quadratic.

savoy123 · 2008-01-12 21:54:35

Hello everyone.

I am looking for a suggestion on a way I could estimate a length of a parabola. I know it's probably not a pretty good description so here is a picture of the problem.

Imagine that this is a roller coaster track. It costs $100 per metre.
NOTE that sections AB, BC, CD is a parabola. Also, point A is inclined at an angle of 10° in which I'm not sure what it means.
How would I attempt to calculate the approximate cost of the new track? I've thought about using Pythagoras but I think it would be too inaccurate.

Thank you for any suggestions on this challenging question.

Last edited by savoy123 (2008-01-12 21:56:09)

luca-deltodesco · 2008-01-12 22:47:42

im not sure if theres a precise way of doing it or not. but you can always approximate it by sampling the position along the track at discrete intervals along the x-axis and use pythagoras to find length of each chord.

mikau · 2008-01-12 23:07:24

i imagine they are showing you this only as a prep for how its done in calculus, namely, sum the distances from A to B, B to C and C to D, thats a simple answer. Judging from the picture, one might argue that if we mark M as the intersection of the lines passing through A-B and C-D respectively, then i think we'd all agree that A to B to M to C to D is a better approximation of the distance.

I'm not sure if this is a homework problem, or if you actually want to build a roller coaster but i think we can actually solve this exactly using fairly simple calculus.

suppose the origin is centered at the bottom left of that diagram. Consider the left most parabolla, it is of the form ax^2 + bx + c = y; note we can find c = 30 just from the origin, for the remaining two variables we have a slope value at A, 2ax + b = tan(10 degrees) at x = 0 thus b = tan(10 deg). a sample point at B then allows us to determine the value of a = -1/8 tan(11) - 15/32. thus we can find the formula for that parabolla and by integrating with the help of obnoxious trig substitutions we can find the length of the portion from 0 to 8.

now we can figure out the rest of the length if we are willing to make one assumption and that is that the curve is differentiable, a smooth curve. if it is then we know the slope of the next parabolla at the starting point, (at B) and we also have two sample points. Continuing in this manner we should be able to find the exact length.

so we could figure it out. The question is, how bad do you need to know?

Last edited by mikau (2008-01-12 23:23:55)

Identity · 2008-01-13 01:14:56

Uh... this is the formula for the length of a curve:

If you've learnt integration, then

1. Find the equations of each parabola section through simultaneous equations
2. Find the derivatives of each equation
3. Plug the derivatives into the formula and integrate away
4. Sum up the results to find the total length

Last edited by Identity (2008-01-13 01:17:17)

luca-deltodesco · 2008-01-13 01:39:09

i was thinking of how you would get an exact equation from integration through using pythagoras' theorem with the derivitive, i just thought it was too silly, but i guess not

Identity · 2008-01-13 01:44:39

luca-deltodesco wrote:

i was thinking of how you would get an exact equation from integration through using pythagoras' theorem with the derivitive, i just thought it was too silly, but i guess not

Good intuition luca! Try and see if you can prove it

luca-deltodesco · 2008-01-13 01:54:26

well i doubt this is a rigid enough proof.

but if you approximate via straight lines by pythagoras you'll have

anyways, parabolas

(used u and v instead of 'a' and 'b' since ive used them in parabola equation)

according to wolfram integrator:

so atleast its directly calculatable. so you could program this easily (sub u and v for x and calculate difference for the length between x = u and x = v)

heres a quick flash app. i put together: http://www.helnet.org/ngup/uploads/quadratic-finder-lengther.swf

it uses the three draggable points to solve the 3 quadratic equations through a matrix inversion (it doesnt work if coef. of x² = 0, since matrix is then singular) and then uses the integral to find the length between the blue and red point on curve

treating the 3 points as coordinates:

it solves the equation:

for the quadratic equation y=ax²+bx+c satisfying those 3 sets of values

Last edited by luca-deltodesco (2008-01-13 02:32:41)

savoy123 · 2008-01-13 02:33:51

Hey again everyone.
First of all, thanks for all the help.

Secondly, I forgot to add that the question asked me to find the APPROXIMATE cost to repair the whole track (so following $100 per meter).

mikau wrote:

so we could figure it out. The question is, how bad do you need to know?

I wasn't worried about the answer but simply developing a way to find the approximate cost of the roller coaster track.

I have done derivatives and such but note that I'm doing maths 1 year ahead of me.

Some posted methods seem quite complicated but I'll try work through them.

Thanks again for all the help and further suggestions are appreciated.

luca-deltodesco · 2008-01-13 02:39:35

my method gives an exact answer why approximate ;P

Last edited by luca-deltodesco (2008-01-13 02:40:21)

mikau · 2008-01-13 08:51:18

I wasn't worried about the answer but simply developing a way to find the approximate cost of the roller coaster track.

if you are expected to know derivatives, then you can find the equations of the parabollas. Once you know that, you can select a handful of points on the parabolla, connect the points and use pythagoras to find the lengths. The next best thing is pretty much just to solve it exactly.

mathsyperson · 2008-01-13 10:57:57

The problem with integrating is that you need to know the function of the curve.
As far as I can tell, the only things you know about the coaster are the co-ordinates of A, B, C and D, and the fact that there's a minimum between B and C.

That would mean an exact answer isn't possible, and the best you could do is one of the approximations that mikau mentioned.

mikau · 2008-01-13 11:15:37

yeah, but we also know that these are parabollas! thats a big deal because we can exactly determine the shape. Our only assumption is that the slope function be continuous, no sharp turns at the joint. Since this is intended to be a smooth track and since it passes through all the points listed, i'd say this assumption is almost certainly correct and if not, only off by a very small margin.

Identity · 2008-01-13 12:36:31

mathsyperson wrote:

The problem with integrating is that you need to know the function of the curve.
As far as I can tell, the only things you know about the coaster are the co-ordinates of A, B, C and D, and the fact that there's a minimum between B and C.
That would mean an exact answer isn't possible, and the best you could do is one of the approximations that mikau mentioned.

I think D and A are maximums... if they aren't, then at least using them as maximums you'd get a very good approximation

Math Is Fun Forum

#1 2008-01-12 21:54:35

NOT Looking for an Answer - Length of Quadratic.

#2 2008-01-12 22:47:42

Re: NOT Looking for an Answer - Length of Quadratic.

#3 2008-01-12 23:07:24

Re: NOT Looking for an Answer - Length of Quadratic.

#4 2008-01-13 01:14:56

Re: NOT Looking for an Answer - Length of Quadratic.

#5 2008-01-13 01:39:09

Re: NOT Looking for an Answer - Length of Quadratic.

#6 2008-01-13 01:44:39

Re: NOT Looking for an Answer - Length of Quadratic.

#7 2008-01-13 01:54:26

Re: NOT Looking for an Answer - Length of Quadratic.

#8 2008-01-13 02:33:51

Re: NOT Looking for an Answer - Length of Quadratic.

#9 2008-01-13 02:39:35

Re: NOT Looking for an Answer - Length of Quadratic.

#10 2008-01-13 08:51:18

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#11 2008-01-13 10:57:57

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#12 2008-01-13 11:15:37

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#13 2008-01-13 12:36:31

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