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Math Help????????????????????
Gravel is being dumped from a conveyor belt at a rate of 35 ft3/min, and its coarseness is such that it forms a pile in the shape of a cone whose base diameter and height are always equal. How fast is the height of the pile increasing when the pile is 10 ft high?
1 Answer
- 8 years agoFavorite Answer
Volume of a cone is = 1/3 * Pi * r² * h
Since diameter = height, then r = h/2
So the volume is = 1/3 * Pi * (h/2)² * h
If the cone is 10' high, then the volume is
1/3 * Pi * (10/2)² * 10 = 250/3 * Pi ft³
At that moment, the volume is increasing by 35 ft³ / minute, so if we add 35 ft³ to the volume of the cone, we'll be able to determine the increase in height over the same minute.
250/3 * Pi + 35 = 1/3 * Pi * (h/2)² * h
250/3 * Pi + 35 = 1/3 * Pi * h³/4
250 * Pi + 105 = Pi * h³/4
250 + 105/Pi = h³/4
1000 + 420/Pi = h³
(1000 + 420/Pi)^(1/3) = h
h = 10,42713
So during that minute, the height will have increased from 10' to 10,42713 feet, an increase of 0.42713 ft/minute
