Sand of Time

Calculus Level 4

An hour glass is constructed from rotating the graphs \( y = \frac{x^{2}}{3}\) and \( y = -\frac{x^{2}}{3}\) around the y-axis within the domain of \([-6, 6]\) as shown in the picture above.

The upper inverted bulb of the hour glass has a circular base with radius of \(6\) cm. and is \(12\) cm. high, and the sand, filled in the upper bulb, is dripping into the lower bulb at a constant rate, falling all down in an hour.

What is the rate of decrease in height of the upper bulb in cm./minute when half an hour has passed?

Give your answer to the nearest 3 decimal places.


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