The rate of change of the volume of water is 0.8π cubic meters per minute.
Given:
Height of the conical tank (h) = 3m
Radius of the conical tank at the top (r) = 2m
Rate of change of water level (dh/dt) = 0.3m/min
Water level (h) = 2m
We need to find the rate of change of the volume of water (dV/dt).
The volume of a cone is given by:
V = (1/3)πr²h
We can differentiate both sides of the equation to find the rate of change of the volume of water:
dV/dt = (1/3)πr²dh/dt
Plugging in the given values:
dV/dt = (1/3)π(2m)²(0.3m/min) = 0.8π m³/min
Therefore, the rate of change of the volume of water is 0.8π cubic meters per minute.