Gravel is being dumped from a conveyor belt at a rate of 50 cubic feet per minute. It forms a pile in the shape of a right circular cone whose base diameter and height are always the same. How fast is the height of the pile increasing when the pile is 22 feet high? Recall that the volume of a right circular cone with height h and radius of the base r is given by V=13πr2h. 

When the pile is 22 feet high, its height is increasing at  feet per minute.

To determine how fast the height of the gravel pile is increasing when the pile is 22 feet high, we start by using the given information and applying related rates.

Given:
- The volume of a right circular cone: \( V = \frac{1}{3} \pi r^2 h \)
- The rate at which gravel is being dumped: \( \frac{dV}{dt} = 50 \, \text{ft}^3/\text{min} \)
- The height of the pile: \( h = 22 \, \text{ft} \)
- The base diameter and height are always the same, so \( r = \frac{h}{2} \)

First, substitute \( r = \frac{h}{2} \) into the volume formula:
\[ V = \frac{1}{3} \pi \left( \frac{h}{2} \right)^2 h = \frac{1}{3} \pi \frac{h^2}{4} h = \frac{1}{12} \pi h^3 \]

Now, differentiate both sides with respect to time \( t \):
\[ \frac{dV}{dt} = \frac{1}{12} \pi \cdot 3h^2 \cdot \frac{dh}{dt} = \frac{1}{4} \pi h^2 \frac{dh}{dt} \]

We are given \( \frac{dV}{dt} = 50 \, \text{ft}^3/\text{min} \) and \( h = 22 \, \text{ft} \). Substitute these values into the equation:
\[ 50 = \frac{1}{4} \pi (22)^2 \frac{dh}{dt} \]
\[ 50 = \frac{1}{4} \pi \cdot 484 \frac{dh}{dt} \]
\[ 50 = 121 \pi \frac{dh}{dt} \]
\[ \frac{dh}{dt} = \frac{50}{121 \pi} \]

Calculate the numerical value:
\[ \frac{dh}{dt} \approx \frac{50}{380.1327} \]
\[ \frac{dh}{dt} \approx 0.1315 \, \text{ft/min} \]

Therefore, the height of the pile is increasing at approximately \( 0.1315 \, \text{feet per minute} \) when the pile is 22 feet high.

Would you like more details on any step or have any other questions? Here are some additional questions you might find interesting:

1. How would the rate change if the dumping rate doubled?
2. What is the radius of the pile when its height is 30 feet?
3. How would the volume formula change if the height were not equal to the diameter?
4. What is the height after 10 minutes of dumping if it started from 0?
5. How does the shape of the pile affect the rate of height increase?

**Tip:** When solving related rates problems, always express all variables in terms of a single variable if possible, to simplify differentiation.

Explore a related rates problem involving a gravel pile forming a right circular cone. Calculate how fast the height of the pile increases when it reaches 22 feet. Detailed step-by-step solution included.

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