Answer:
-25/(12π) ft/min ≈ -0.663 ft/min
Step-by-step explanation:
You want the rate of change of height of the salt in a cone with height 5 ft and diameter 4 ft, filled 3 ft up from the vertex, and decreasing in volume at the rate of 3 cubic feet per minute.
The rate of change of height is the rate of change of volume, divided by the surface area.
The given cone dimensions tell you the radius is 2/5 of the height of the cone, so the radius of the surface when it is filled to a depth of 3 ft is ...
r = (2/5)(3 ft) = 1.2 ft
The area of the surface is the area of a circle with this radius:
A = πr² = π(1.2 ft)² = 1.44π ft²
Then the rate of change of height is ...
h' = V'/A
h' = (-3 ft³/min)/(1.44π ft²) = -25/(12π) ft/min ≈ -0.663 ft/min
