perplexus.info :: Calculus : Roll away

Roll away (Posted on 2018-08-15)

Graph y = x³/3 - x²/2

Imagine a small circle sitting inside the local minimum. Gravity pulls in the negative y direction so it is quite stable there.

Now increase the size of the circle. At some point it will become too large to fit in this hollow and will be forced to roll away down into the third quadrant.

At what radius does this happen?

No Solution Yet Submitted by Jer

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An approximate look by graphing

| Comment 1 of 5

The radius at which it starts to slide down and to the left seems to be about 2.53. It's the size at which the circle sits precisely tangent to the local maximum of the cubic, at the origin, and is tangent to the cubic at another point (as it had been all the way through its growth).

This is based on graphing y=2.53-sqr(2.53^2-x^2) on the same plot as y=x^3/3-x^2/2, after having adjusted both instances of what is now 2.53, so as to get a point of tangency other than the origin.

Posted by Charlie on 2018-08-15 11:50:18

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