Final answer:
To determine dh/dt, we use the volume of a cone formula and differentiate it with respect to time. By substituting in the proportional relationship between radius and height in the conical tank and using the given flow rate, we can solve for dh/dt.
Step-by-step explanation:
To find the rate of change of the height of water in the tank, dh/dt, as a function of time (t), we need to apply the concept of related rates in calculus to the volume of a cone. The volume V of a cone with radius r and height h is given by the formula V = (1/3)πr^2h. Because the conical tank has a fixed shape, the radius and height of the water level are related by the proportions of the tank, so we have r/h = 4/8 = 1/2, or r = (1/2)h.
Differentiating both sides of the volume formula with respect to time t gives us dV/dt = (1/3)π(2rh dr/dt + r^2 dh/dt). Since r is proportional to h, we can substitute r with (1/2)h in the equation and solve for dh/dt using the given flow rate, 2.4 m³/min, which is dV/dt.