Final answer:
To determine the changing rate of the water height in the funnel, we use related rates and calculus, applying the volume formula of a cone and differentiating with respect to time. We then solve for dh/dt using the given drainage rate and the relevant dimensions of the funnel.
Step-by-step explanation:
To find the rate at which the height of the water in the funnel is changing when the water level is at a height of 2 ft, we must use calculus and the concept of related rates. The volume of water in the funnel can be described by the formula for the volume of a cone, which is V = (1/3)πr^2h, where V is volume, r is radius, and h is height. Since the funnel tapers down uniformly, the ratio of the height to the radius remains constant, which means that r/h = R/H where R is the known radius of 5ft at the height H of 8ft.
We are given that the volume of water V is decreasing at a rate of 0.4 ft³/sec, so we have dV/dt = -0.4 ft³/sec. Using the chain rule, we can differentiate both sides of the volume formula with respect to time (t) to find the relationship between the rates of change of the volume, radius, and height. Through substitution and solving for dh/dt, we'll find the rate at which the height is changing when h is 2ft.