Answer:
-4.697 in/s
Step-by-step explanation:
You want to know the rate of change of radius of a cone 11 inches high with a radius of 5 inches and a volume that is decreasing at the rate of 541 cubic inches per second.
<h3>Volume</h3>
The volume of a cone is given by ...
V = π/3r²h
The rate of change is found by implicit differentiation:
V' = (π/3)((2rr')h +r²h')
Here, the height is constant, so h' = 0. Solving for r', we find ...
r' = 3V'/(2πrh)
<h3>Application</h3>
Using the given values of V', r, and h, we find the rate of change of radius to be ...
r' = 3(-541 in³/s)/(2π(5 in)(11 in)) = -1623/(110π) in/s ≈ -4.69652 in/s
The radius is decreasing at about -4.697 inches per second.
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<em>Additional comment</em>
The initial radius of the cone is irrelevant to the problem, since we want to know the rate of change at a specific different radius. Whether the cone is inverted or not is also irrelevant to its volume.
