A cycloid is the curve formed by a fixed point on the edge of a wheel as the wheel rotates along the ground. The curve is given by the parametric equations
x = r (t - sin t),
y = r (1 - cos t)
where r is the radius of the wheel. If you take one arch of the graph and turn it upside down it forms a ramp with a special property: it is the fastest path for a ball rolling down from one point to another. A second interesting fact: if you do have a ramp in this shape, no matter where you start a ball rolling, it will always take the same amount of time to reach the bottom. The cool photo is from illum on Flickr.
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Related links:
Wikipedia entry for cycloid
YouTube video of a marble rolling down a cycloid ramp
It turns out that the force exerted on the dam doesn't depend on how much water is in the lake, but rather just on the depth of the water. It is tricky to calculate the total force on a dam however because the water pressure is different at different depths. So the bottom of the dam will experience a lot more force than the top. You can estimate the total force by considering horizontal sections of the dam separately, and if you want to be more accurate, calculus can get you to the precise value.