Hello All and Happidy New Year. I am setting here sipping on some Tangueray Rangpur and thought I would pose a rather fun calc applications problem.

"An amusement park ride has cars which move along a track (I hope I can get it across in my haphazard diagram). The cars are attached to arms that are 20 feet long and are connected to a common point on the axis of rotation(z-axis). The arms swing up and down along the track whose elevation is given by $\displaystyle z=4sin^{2}({\theta})$. Assuming all distances are measured in feet, find the length of the track".

I tried drawing the picture, but a I think I got it a little askew. Not too bad, though. It's height is 0 where it touches the x-axis and at the points over the y-axis it is 4 feet in elevation. The circumference of a circle with radius 20 is 125.66, so we know it'll be a wee bit more than that.

If anyone wants to try it, see what you come up with. I attempted it in both polar and spherical and arrived at the same answer using the arc length formulae. So, I am pretty sure I am correct.

I used @ for theta because that's easier in Paint.