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Hello! Let's work through the problem you've provided. 

### Problem Overview:
You are given a funnel where the radius \( r \) at height \( h \) is defined by the function:
\[
r = \frac{1}{20} (3 + h^2)
\]
with \( 0 \leq h \leq 10 \), and \( r \) and \( h \) are both measured in inches.

The problem asks for the rate of change of the height \( \frac{dh}{dt} \) when the height \( h = 3 \) inches, and the radius \( r \) is decreasing at a rate of \( \frac{dr}{dt} = -\frac{1}{5} \) inches per second.

### Solution:
We need to find \( \frac{dh}{dt} \) using the relationship between \( r \) and \( h \). Since \( r \) is a function of \( h \), we can differentiate the function with respect to time \( t \).

#### 1. Differentiate the given function with respect to \( t \):
\[
\frac{d}{dt} \left( r = \frac{1}{20} (3 + h^2) \right)
\]
Applying the chain rule:
\[
\frac{dr}{dt} = \frac{1}{20} \cdot \frac{d}{dt} (3 + h^2) = \frac{1}{20} \cdot 2h \cdot \frac{dh}{dt}
\]
Simplifying:
\[
\frac{dr}{dt} = \frac{h}{10} \cdot \frac{dh}{dt}
\]

#### 2. Substitute the given values:
At \( h = 3 \) inches and \( \frac{dr}{dt} = -\frac{1}{5} \) inches per second:
\[
-\frac{1}{5} = \frac{3}{10} \cdot \frac{dh}{dt}
\]

#### 3. Solve for \( \frac{dh}{dt} \):
Multiply both sides by \( \frac{10}{3} \) to isolate \( \frac{dh}{dt} \):
\[
\frac{dh}{dt} = -\frac{1}{5} \cdot \frac{10}{3} = -\frac{2}{3} \text{ inches per second}
\]

### Final Answer:
The rate of change of the height of the liquid with respect to time when \( h = 3 \) inches is \( \frac{dh}{dt} = -\frac{2}{3} \) inches per second.

Would you like any further details on this solution, or do you have any additional questions?

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Here are 5 related questions you might find useful:
1. How would the problem change if the radius was increasing instead of decreasing?
2. What would happen to \( \frac{dh}{dt} \) if the height \( h \) was different, say \( h = 5 \) inches?
3. Can you determine the volume of liquid in the funnel as a function of height \( h \)?
4. What is the significance of the negative sign in \( \frac{dh}{dt} \)?
5. How does the geometry of the funnel affect the relationship between \( r \) and \( h \)?

**Tip:** When dealing with related rates problems, always remember to check the units and signs (positive or negative) of the rates you are calculating to ensure they make sense in the context of the problem.

Learn how to solve a related rates problem involving a funnel. Find the rate of change of height when the radius is decreasing at a specified rate. Detailed step-by-step solution included.

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