finding lengths of nonlinear curves (length of track)

[Iamadam]

I'm having real troubles with the following question:

I know that line AB and BC can be modelled using quadratic equations, and CD is just a linear equation. I just have no idea how to find the equations. Any help would be appreciated.

So far, I have only tried to find AB (it seems to be a logical place to start). I have rotated the model 90*, and assumed that 'B' is the turning point of the equation.

. . .f`(20) = 2a(20) + b

I get lost around there. I think I'm looking at this whole thing wrong, so I decided to write a scaled down problem. I have got three equations. Two are quadratics, one is linear.

My question is: How can I find the lengths of the lines created by these functions, over a given domain? (I know how to do the linear one.) For example (just off the top of my head), given this line:

. . .y = 2x^2 + 5x + 6

...how would I find the length of the line over the interval 0 < x < 3

Thank you!

 

[galactus]

This is a parabola. If I understand correctly, you want to know how to find the arc length along the parabola 2x^2+5x+6 from 0 to 3?.

\(\displaystyle \L\\\int_{0}^{3}\sqrt{1+(\frac{d}{dx}[2x^{2}+5x+6])^{2}}dx=33.152\)

To find the length of a straight line between those points, just use the x and y values. x=0 and 3. Plugging those into the function gives y=39 and 6.

Now, use Pythagoras: \(\displaystyle \L\\\sqrt{(3-0)^{2}+(39-6)^{2}}=3\sqrt{122}\approx{33.136}\)

Note the length along the parabola is slightly longer than the line because of the arc.