A conical salt spreader is spreading salt at a rate of 2 cubic feet per minute. The diameter of the base of the cone is 4 feet and the height of the cone is 5 feet.
How fast is the height of the salt in the spreader decreasing when the height of the salt in the spreader (measured from the vertex of the cone upward) is 3 feet? Give your answer in feet per minute.
How fast is the height of the salt in the spreader decreasing when the height of the salt in the spreader (measured from the vertex of the cone upward) is 3 feet? Give your answer in feet per minute.
-
dV/dt = -2
V = ⅓ πr²h
relationship between radius and height
r / h = 2 / 5
r = ⅖ h
`````````
V = ⅓ πr²h
V = ⅓ π( ⅖ h)²h
V = 4πh³ / 75
````````````````````
V = 4πh³ / 75
differentiate with respect to time, t
V = 4πh³ / 75
V' = (4πh² / 25) h'
h' = 25V' / (4πh²)
h'(3) = 25(-2) / (4π*3²)
h'(3) = -25 / (18π) ≈ - 0.442 ft/min
V = ⅓ πr²h
relationship between radius and height
r / h = 2 / 5
r = ⅖ h
`````````
V = ⅓ πr²h
V = ⅓ π( ⅖ h)²h
V = 4πh³ / 75
````````````````````
V = 4πh³ / 75
differentiate with respect to time, t
V = 4πh³ / 75
V' = (4πh² / 25) h'
h' = 25V' / (4πh²)
h'(3) = 25(-2) / (4π*3²)
h'(3) = -25 / (18π) ≈ - 0.442 ft/min