Mathematics

Problem

Water drains from the bottom of an inverted right circular cone at a constant rate of

12\pi\,\text{cm}^3\,\text{s}^{-1}

. The cone itself has fixed height

30\,\text{cm}

and base radius

15\,\text{cm}

. At the instant when the water depth is

9\,\text{cm}

, find the rate, in cm s⁻¹, at which the depth of the water is decreasing.

Intuitive explanation

The key is to link the volume draining away to how fast the water level drops using similar triangles. Since the cone is inverted, the water always forms a smaller cone whose radius and height stay proportional to the full cone's dimensions—use this ratio to express volume in terms of depth alone, then differentiate with respect to time to connect $\frac{dV}{dt}$ to $\frac{dh}{dt}$ .

Intuitive explanation

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