Gravel is being dumped from a conveyor belt at a rate of 30 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 15 ft high?
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To find the rate at which the height of the pile is increasing, we can use related rates in calculus.
Let's denote the height of the pile as "h" and the radius/base diameter as "r". Since the pile is in the shape of a cone, we know that the radius is equal to half the base diameter, so r = h/2.
The volume of a cone is given by the formula V = (1/3)πr²h, where π is pi. We are given that gravel is being dumped from a conveyor belt at a rate of 30 ft³/min, so the rate at which the volume is increasing is dV/dt = 30 ft³/min.
Using the formula for the volume of a cone and differentiating with respect to time (t), we can obtain an equation that relates the rates of change of volume (dV/dt), height (dh/dt), and radius (dr/dt):
dV/dt = (1/3)(πr²)(dh/dt) + (2/3)(πr)(dr/dt)
Substituting the given values, we can further simplify the equation:
30 = (1/3)(π(h/2)²)(dh/dt) + (2/3)(π(h/2))(dr/dt)
Now, we need to solve for dh/dt, which represents the rate at which the height is changing with respect to time. We know that the height of the pile is 15 ft when we want to find the rate of change, so we substitute h = 15:
30 = (1/3)(π(15/2)²)(dh/dt) + (2/3)(π(15/2))(dr/dt)
Now we can solve for dh/dt:
dh/dt = [30 - (2/3)(π(15/2))(dr/dt)] / [(1/3)(π(15/2)²)]
To find dr/dt, the rate at which the radius is changing, we need to relate it to dh/dt. The radius and height are related as r = h/2, so dr/dt = (1/2)(dh/dt).
Substituting this value, we have:
dh/dt = [30 - (2/3)(π(15/2))(1/2)(dh/dt)] / [(1/3)(π(15/2)²)]
To further simplify the equation, we multiply the denominators through:
dh/dt = [3(30 - (2/3)(π(15/2))(1/2)(dh/dt))] / [π(15/2)²]
Simplifying the numerators:
dh/dt = [90 - (π/6)(15)(dh/dt)] / [π(15/2)²]
Now, we can isolate dh/dt:
π(15/2)²(dh/dt) = 90 - (π/6)(15)(dh/dt)
Multiplying out the terms:
(π(15/2)² + (π/6)(15))(dh/dt) = 90
(π/4)(15)²(dh/dt) = 90
dh/dt = 90 / [(π/4)(15)²]
Finally, we can calculate the value of dh/dt:
dh/dt ≈ 0.152 ft/min