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The question you're working on involves a related rates problem where sand is falling onto a conical pile.

### Key Information from the Problem:
- Sand is falling at a rate of \( 16 \, \text{cubic feet per minute} \), i.e., \( \frac{dV}{dt} = 16 \, \text{ft}^3/\text{min} \).
- The diameter of the base of the cone is three times the height: \( d = 3h \), which implies \( r = \frac{d}{2} = \frac{3h}{2} \).
- The height of the pile is \( 22 \, \text{feet} \).
- The formula for the volume of a cone is: 
  \[
  V = \frac{1}{3} \pi r^2 h
  \]
- We need to find \( \frac{dh}{dt} \) when \( h = 22 \, \text{feet} \).

### Solution Outline:
1. **Volume Equation Substitution:**
   From the relation \( r = \frac{3h}{2} \), substitute into the volume formula:
   \[
   V = \frac{1}{3} \pi \left( \frac{3h}{2} \right)^2 h = \frac{1}{3} \pi \left( \frac{9h^2}{4} \right) h = \frac{3\pi}{4} h^3
   \]
   So, the volume in terms of \( h \) is:
   \[
   V = \frac{3\pi}{4} h^3
   \]

2. **Differentiate with respect to time \( t \):**
   Use implicit differentiation to find the rate of change of volume with respect to time:
   \[
   \frac{dV}{dt} = \frac{3\pi}{4} \cdot 3h^2 \frac{dh}{dt} = \frac{9\pi}{4} h^2 \frac{dh}{dt}
   \]

3. **Solve for \( \frac{dh}{dt} \):**
   Since \( \frac{dV}{dt} = 16 \, \text{ft}^3/\text{min} \), plug that into the equation:
   \[
   16 = \frac{9\pi}{4} h^2 \frac{dh}{dt}
   \]
   When \( h = 22 \, \text{feet} \), substitute this value:
   \[
   16 = \frac{9\pi}{4} (22)^2 \frac{dh}{dt}
   \]
   Simplify:
   \[
   16 = \frac{9\pi}{4} \times 484 \times \frac{dh}{dt}
   \]
   \[
   16 = 1089\pi \frac{dh}{dt}
   \]
   Solve for \( \frac{dh}{dt} \):
   \[
   \frac{dh}{dt} = \frac{16}{1089\pi}
   \]

4. **Numerical Solution:**
   Approximating the value of \( \pi \):
   \[
   \frac{dh}{dt} \approx \frac{16}{1089 \times 3.1416} \approx \frac{16}{3421.97} \approx 0.00467 \, \text{ft/min}
   \]

Thus, the rate at which the height of the pile is changing is approximately \( 0.00467 \, \text{ft/min} \).

Would you like more details on any step or have any other questions?

Here are five related questions:
1. How would the solution change if the diameter was twice the height instead of three times?
2. What if the rate of sand falling was not constant but a function of time?
3. Can you calculate the rate of change of the radius when the height is 22 feet?
4. How would this problem change if it involved a pyramid instead of a cone?
5. What if the base of the cone was not circular but elliptical?

**Tip:** Always double-check units and make sure they are consistent throughout your calculations to avoid errors.

At a sand and gravel plant, sand is falling off a conveyor and onto a conical pile at a rate of 16 cubic feet per minute. The diameter of the base of the cone is approximately three times the altitude. At what rate (in ft/min) is the height of the pile changing when the pile is 22 feet high?

This problem involves finding the rate at which the height of a sand pile is changing as sand falls at a rate of 16 cubic feet per minute. Using related rates and cone geometry, this problem is suitable for undergraduate calculus students.

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