At a quarry, sand is filling off a conveyor onto a conical pile at a rate of 15 cubic feet per minute. The diameter of the base of the cone is four times the height. At what rate is the height of the pile changing when the pile is 12 feet high?
To find the rate at which the height of the pile is changing, we need to use related rates. Let's break down the problem into smaller parts and variables:
Let:
V = Volume of the cone
h = Height of the pile
r = Radius of the base of the cone
We are given:
dV/dt = 15 cubic feet per minute (the rate at which the sand is filling onto the pile)
The diameter of the base is four times the height (d = 4h)
We can start by relating the variables using the formula for the volume of a cone:
V = (1/3) * π * r^2 * h
Since the diameter of the base is four times the height, we know that the radius is half the diameter:
r = d/2 = (4h)/2 = 2h
Now we can express the volume of the cone only in terms of h:
V = (1/3) * π * (2h)^2 * h
V = (8/3) * π * h^3
To find dh/dt, which represents the rate at which the height of the pile is changing when the pile is 12 feet high, we need to differentiate both sides of the equation with respect to time (t):
dV/dt = d((8/3) * π * h^3)/dt
Using the chain rule, this becomes:
15 = (8/3) * π * 3h^2 * dh/dt
Simplifying:
15 = (8/3) * π * 3h^2 * dh/dt
15 = 8πh^2 * dh/dt
Now, we can solve for dh/dt when h = 12:
15 = 8π(12)^2 * dh/dt
We can rearrange the equation to solve for dh/dt:
dh/dt = 15 / (8π(12)^2)
dh/dt ≈ 0.019 cubic feet per minute
Therefore, the height of the pile is changing at a rate of approximately 0.019 cubic feet per minute when the pile is 12 feet high.
To solve this problem, we can use the related rates method. Let's denote the height of the pile as h and the radius of the base as r.
We are given that the rate of change of the volume of the pile is 15 cubic feet per minute. We know that the volume of a cone is given by the formula: V = (1/3) * π * r^2 * h.
Differentiating the formula with respect to time, we get:
dV/dt = (1/3) * π * (2r * dr/dt * h + r^2 * dh/dt)
Since the base diameter is four times the height, we can write r = 2h.
Substituting this into the equation and substituting the known values, we have:
15 = (1/3) * π * (2(2h) * dr/dt * h + (2h)^2 * dh/dt)
Simplifying:
15 = (1/3) * π * (4h * dr/dt * h + 4h^2 * dh/dt)
Now, we need to solve for dh/dt, the rate at which the height of the pile is changing when the pile is 12 feet high.
Let's substitute h = 12 into the equation:
15 = (1/3) * π * (4(12) * dr/dt * 12 + 4(12)^2 * dh/dt)
Simplifying:
15 = (16/3) * π * (12 * dr/dt + 144 * dh/dt)
Dividing both sides by (16/3) * π * (12 * dr/dt + 144 * dh/dt):
15 / ((16/3) * π * (12 * dr/dt + 144 * dh/dt)) = 1
To find the value of dh/dt, we need to solve this equation. However, we don't have the value for dr/dt.