Gravel is being dumped from a conveyor belt at a rate of 50 cubic feet per minute. It forms a pile in the shape of a right circular cone whose base diameter and height are always equal to each other. How fast is the height of the pile increasing when the pile is 14 feet high?
Hint: The volume of a right circular cone with height h and radius of the base r is given by V = [1/3] πr2 h.
40/121pi
To find the rate at which the height of the pile is increasing, we need to use related rates and the information given in the problem.
Let's use the variables:
- V: Volume of the pile (in cubic feet)
- h: Height of the pile (in feet)
- r: Radius of the base of the pile (in feet)
We know that the rate at which gravel is being dumped from the conveyor belt is 50 cubic feet per minute. Therefore, we can express the rate of change of volume with respect to time:
dV/dt = 50 ft³/min
The volume of a right circular cone is given by:
V = (1/3) * π * r² * h
Since the problem states that the base diameter and height are always equal, we have:
d = 2r
or
r = d/2
We are given the height of the pile (h = 14 feet) and need to find the rate at which it is changing. We can express the height of the pile in terms of r using similar triangles:
h/r = H/R
Where H is the height of the cone and R is the radius of the base. Since the diameter is equal to the height, we can substitute h for R:
h/r = H/h
H = (h²)/r
Now, let's differentiate the volume equation with respect to time (t):
dV/dt = (1/3) * π * (2r * dr/dt * h + r² * dh/dt)
We can simplify this equation by substituting the expression we derived for H:
dV/dt = (1/3) * π * (2r * dr/dt * h + r² * dh/dt)
50 = (1/3) * π * (2r * dr/dt * h + r² * dh/dt)
Now, we have an equation in terms of known values (r, h, dr/dt) and the unknown value (dh/dt). We need to substitute the known values and solve for dh/dt.
Given:
r = d/2
h = 14 ft
dr/dt = ?
We need to find dh/dt when the height of the pile is 14 ft, which means:
h = 14 ft
To find dr/dt, we need information about how fast the radius is changing. Unfortunately, the problem does not provide this information. Without the rate at which the radius is changing, we cannot determine the rate at which the height is changing.
Therefore, we cannot find the rate at which the height of the pile is increasing without additional information.