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Rationalize the denominator:
\]Question: A weather balloon is modeled as a sphere with radius $ r $ meters, and it ascends such that its volume increases at a rate proportional to $ \sin \theta $, where $ \theta $ is the angle of elevation from the ground. If the volume increase rate is given by $ \frac{dV}{dt} = k \sin \theta $, and $ \theta = 60^\circ $, compute the rate of volume increase in terms of $ k $ and $ r $.
Solution: The volume of a sphere is given by $ V = \frac{4}{3} \pi r^3 $. Differentiating with respect to time $ t $, we get:

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