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Water is poured into an inverted right circular cone. The cone has a height of 12 cm and a radius of 6 cm at the top. The water is being poured in at a constant rate of 4 cm\(^3\) per second. Let \(r\) be the radius of the water's surface and \(h\) be the height of the water at time \(t\).
Show that the radius of the water's surface is always half of its height, i.e., \(r = \frac{h}{2}\).
Find the rate at which the height of the water is increasing when the water is 8 cm deep.
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