Polar-coordinate Problem

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I don't even know where to begin with this problem, can anybody help?
If I were to do this problem, I would first use wolframalpha.com to plot:

r = f(Θ) = Θ^3 - 6*Θ^2 + 9*Θ

That will let you analyze the position of the particle.

If you get stuck somewhere, come back and tell us exactly where you are stuck....
 
Hi natH. The distance from the Origin to a point on the polar curve is r. We're told that r-values are the same as f-values. We're told that t is the same number as θ. Therefore, distance r is a function of time:

r(t) = t^3 - 6t^2 + 9t

Do you know how to use calculus, to find the intervals where distance r(t) is increasing, while 0 ≤ t ≤ 2pi seconds?

?
 
… is D the correct answer?
Yes. A function increases when its first derivative (rate-of-change, or slope) is positive.

The first derivative of r is an easily-factored quadratic polynomial.

So, it's easy to determine intervals where the parabola lies above the horizontal axis. That serves as a check, for answer D.

?
 
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