Final answer:
The rate of change of the height of the cone is approximately -0.314 feet per second at the instant when the height is 66 feet.
Step-by-step explanation:
Given that the radius of the cone is decreasing at a constant rate of 5 feet per second and the volume remains constant at 432 cubic feet, we can use the formula for the volume of a cone to find the relationship between the radius and height of the cone. The formula for the volume of a cone is V = (1/3) * π * r^2 * h, where V is the volume, r is the radius, and h is the height. By plugging in the given volume and the current radius, we can solve for the current height of the cone. Once we have the current height, we can differentiate the volume formula with respect to time to find the rate of change of the height.
Using the given volume of 432 cubic feet and the current radius, we can solve for the current height:
V = (1/3) * π * r^2 * h
432 = (1/3) * π * 66^2 * h
h = 432 * 3 / (π * 66^2)
h ≈ 6.12 feet
Now, we can differentiate the volume formula with respect to time:
dV/dt = (1/3) * π * (2rh * dr/dt + r^2 * dh/dt)
Since the volume is constant, dV/dt = 0. We can plug in the known values: dh/dt = ?, r = 66 feet, and dr/dt = -5 feet/second:
0 = (1/3) * π * (2 * 66 * -5 + 66^2 * dh/dt)
solving for dh/dt, the rate of change of the height:
dh/dt = 2 * 66 * -5 / (66^2 * π) = -660 / (66^2 * π) ≈ -0.314 feet/second