Mathematics

Problem

A right circular cone with height

10\ \text{m}

and base radius

4\ \text{m}

is filled with water. The water depth

h

decreases at a constant rate of

0.20\ \text{m/min}

. When

h=6\ \text{m}

, determine the instantaneous rate at which the volume of water in the cone is changing. State your answer in cubic meters per minute.

Intuitive explanation

The key is recognizing that as water drains, the water surface shrinks—so you can't just use the cone's full dimensions. Set up a relationship between the water's radius and depth using similar triangles (the water always forms a cone similar to the original), then differentiate the volume formula with respect to time. This converts a static geometry problem into a related rates problem where you can use the given rate $\frac{dh}{dt}$ to find $\frac{dV}{dt}$ .

Intuitive explanation

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