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.