We are given that sand is falling onto a conical pile at a rate of $\frac{dV}{dt} = 8 \, \text{ft}^3/\text{min}$, where $V$ is the volume of the cone. We are asked to find the rate at which the height $h$ of the cone is changing when the height is 2 feet, given that the diameter of the base is three times the height.

Step 1: Formula for the volume of a cone

The volume $V$ of a cone is given by: $V = \frac{1}{3} \pi r^2 h$ where $r$ is the radius of the base and $h$ is the height.

Step 2: Relate the radius to the height

We are told that the diameter of the base is three times the height, so: $2r = 3h \quad \text{or} \quad r = \frac{3}{2} h$

Step 3: Substitute $r$ into the volume formula

Substituting $r = \frac{3}{2}h$ into the volume formula: $V = \frac{1}{3} \pi \left( \frac{3}{2}h \right)^2 h$ $V = \frac{1}{3} \pi \times \frac{9}{4} h^2 \times h$ $V = \frac{3\pi}{4} h^3$

Step 4: Differentiate the volume with respect to time

To find the rate of change of the height $h$, we differentiate $V = \frac{3\pi}{4} h^3$ with respect to time $t$: $\frac{dV}{dt} = \frac{3\pi}{4} \cdot 3h^2 \frac{dh}{dt}$ $\frac{dV}{dt} = \frac{9\pi}{4} h^2 \frac{dh}{dt}$

Step 5: Substitute known values

We are given $\frac{dV}{dt} = 8 \, \text{ft}^3/\text{min}$ and $h = 2 \, \text{ft}$. Substituting these into the equation: $8 = \frac{9\pi}{4} (2)^2 \frac{dh}{dt}$ $8 = \frac{9\pi}{4} \times 4 \times \frac{dh}{dt}$ $8 = 9\pi \frac{dh}{dt}$

Step 6: Solve for $\frac{dh}{dt}$

$\frac{dh}{dt} = \frac{8}{9\pi}$

Thus, the height of the pile is changing at a rate of $\frac{8}{9\pi} \, \text{ft/min}$, or approximately $0.283 \, \text{ft/min}$.

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Do you need more details or have any questions?

Here are 5 related questions to consider:

1. How would the rate of change of the height vary if the diameter were only twice the height?
2. What happens to the rate of height change if the sand falls at a faster rate (e.g., 10 cubic feet per minute)?
3. How would the problem change if the pile formed a cylinder instead of a cone?
4. Can you derive a general formula for $\frac{dh}{dt}$ based on any relationship between diameter and height?
5. What would happen to the rate of change of height when the pile reaches 10 feet in height?

Tip: When differentiating implicit relationships, always double-check which variables depend on time to avoid missing terms!