To solve this problem, we need to find the rate at which the height of a conical pile is changing, given the rate at which gravel is being added to the pile.

We are given:

• The volume rate of gravel falling on the pile: $\frac{dV}{dt} = 32 \frac{\text{ft}^3}{\text{min}}$
• The relationship between the radius and height of the cone: $r = \frac{1}{2}h$
• The height of the pile at the specific moment: $h = 9.5 \text{ ft}$

First, recall the formula for the volume of a cone: $V = \frac{1}{3} \pi r^2 h$

Substitute $r = \frac{1}{2}h$ into the volume formula: $V = \frac{1}{3} \pi \left(\frac{1}{2}h\right)^2 h = \frac{1}{3} \pi \left(\frac{1}{4}h^2\right) h = \frac{1}{12} \pi h^3$

Now, differentiate both sides with respect to time $t$: $\frac{dV}{dt} = \frac{d}{dt} \left(\frac{1}{12} \pi h^3\right)$

Using the chain rule, we get: $\frac{dV}{dt} = \frac{1}{12} \pi \cdot 3h^2 \cdot \frac{dh}{dt} = \frac{1}{4} \pi h^2 \frac{dh}{dt}$

Given $\frac{dV}{dt} = 32 \frac{\text{ft}^3}{\text{min}}$, substitute this value in: $32 = \frac{1}{4} \pi h^2 \frac{dh}{dt}$

Now, solve for $\frac{dh}{dt}$: $\frac{dh}{dt} = \frac{32}{\frac{1}{4} \pi h^2} = \frac{32 \cdot 4}{\pi h^2} = \frac{128}{\pi h^2}$

Substitute $h = 9.5 \text{ ft}$: $\frac{dh}{dt} = \frac{128}{\pi (9.5)^2} = \frac{128}{\pi \cdot 90.25} = \frac{128}{90.25 \pi} = \frac{128}{\pi} \cdot \frac{1}{90.25}$

Simplify: $\frac{dh}{dt} = \frac{128}{90.25 \pi} = \frac{128 \cdot 4}{361 \pi} = \frac{512}{361 \pi}$

So, the rate at which the height is changing when the pile is 9.5 feet high is: $\frac{dh}{dt} = \frac{512}{361 \pi} \frac{\text{ft}}{\text{min}}$

The correct answer is: $\frac{dh}{dt} = \frac{512}{361 \pi} \frac{\text{ft}}{\text{min}}$

Do you want more details or have any questions?

Here are five questions you might want to ask next:

1. How do you find the volume of a cone?
2. What is the chain rule in differentiation?
3. How do you differentiate a function of the form $V = \frac{1}{12} \pi h^3$?
4. What is the relationship between radius and height in a cone when given a specific ratio?
5. How do you solve for a variable in a given equation?

Tip: When solving related rates problems, always start by writing down the given information and the relationship between the variables. Then, differentiate with respect to time and substitute the known values to solve for the unknown rate.