a conveyor makes a conical pile of mulch which has a height that is 2/3 of the diameter
The conveyor can deliver 10 cubic yards of mulch to the pile per minute. What is the function for the height of the pile as a function of time
v = 1/3 π r^2 h
Now, h = 2/3 d = 4/3 r, so r = 3/4 h
v = 1/3 π (3/4 h)^2 h = 3π/16 h^3
Since v increases at a constant 10 yd^3/min,
v(t) = 10t = 3π/16 h^3
h^3 = 160t/3π
h(t) = 2∛(20t / 3π)
v = h * π * (3/4 h)^2 = 9/16 * π * h^3
v = 10 t
h(t) = [160 t / (9 π)]^(1/3)
forgot the 1/3 in the volume equation
To find the function for the height of the pile as a function of time, let's start by defining some variables:
Let's denote:
h = height of the pile (in yards)
d = diameter of the pile (in yards)
t = time (in minutes)
Given that the height of the pile is 2/3 of the diameter, we can express h in terms of d as:
h = (2/3) * d
Next, we need to relate the diameter of the pile to the time it takes for the conveyor to deliver mulch. We're given that the conveyor delivers 10 cubic yards of mulch per minute. Since the pile is conical, we can use the formula for the volume of a cone to relate the diameter of the pile to the time.
The volume of a cone (V) can be calculated using the formula:
V = (π/3) * r² * h
Since the diameter (d) is twice the radius (r), we have:
r = d/2
Substituting the value for r in the volume formula, we get:
V = (π/3) * (d/2)² * h
The conveyor delivers 10 cubic yards per minute, so the volume of mulch delivered (V) is given by:
V = 10t
Now we can equate the volume of mulch delivered (V) to the volume of the conical pile (V) and solve the equation for h:
(π/3) * (d/2)² * h = 10t
Simplifying the equation:
(π/12) * d² * h = 10t
Substituting h = (2/3) * d:
(π/12) * d² * (2/3) * d = 10t
Simplifying further:
(π/18) * d³ = 10t
Dividing both sides by (π/18):
d³ = (10t * 18) / π
Taking the cube root of both sides:
d = ((10t * 18) / π)^(1/3)
Finally, substitute the expression for d in terms of t back into the equation for h:
h = (2/3) * d
h = (2/3) * ((10t * 18) / π)^(1/3)
Therefore, the function for the height of the pile as a function of time is:
h(t) = (2/3) * ((10t * 18) / π)^(1/3)