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To solve this problem, we need to use the relationship between the volume of the cylinder, its radius \( r \), and height \( h \), along with the information about the rate at which the height is increasing.

### Step 1: Set up the Volume Equation
The volume \( V \) of a right circular cylinder is given by:
\[
V = \pi r^2 h
\]
Since no clay is lost, the volume remains constant over time.

### Step 2: Differentiate with Respect to Time
Differentiate both sides of the equation with respect to time \( t \):
\[
\frac{dV}{dt} = \pi \left( 2r \frac{dr}{dt} \cdot h + r^2 \frac{dh}{dt} \right)
\]
Since the volume is constant, \( \frac{dV}{dt} = 0 \), so we can set the equation to zero:
\[
0 = 2\pi r h \frac{dr}{dt} + \pi r^2 \frac{dh}{dt}
\]

### Step 3: Solve for \( \frac{dr}{dt} \)
Rearrange to isolate \( \frac{dr}{dt} \):
\[
2rh \frac{dr}{dt} = -r^2 \frac{dh}{dt}
\]
\[
\frac{dr}{dt} = -\frac{r}{2h} \frac{dh}{dt}
\]

### Step 4: Substitute Values
We are given:
- \( r = 7 \) cm
- \( h = 10 \) cm
- \( \frac{dh}{dt} = 0.4 \) cm/s

Substitute these values into the equation:
\[
\frac{dr}{dt} = -\frac{7}{2 \cdot 10} \cdot 0.4
\]
\[
\frac{dr}{dt} = -\frac{7 \cdot 0.4}{20}
\]
\[
\frac{dr}{dt} = -0.14 \, \text{cm/s}
\]

### Final Answer
The rate at which the radius is changing when the radius is 7 cm and the height is 10 cm is \( \frac{dr}{dt} = -0.14 \) cm/s.

Would you like a more detailed explanation of each step or have any questions?

### Related Questions
1. How would the rate of change in radius be affected if the height increase rate changed?
2. What happens to the radius if the volume is allowed to change?
3. How would this calculation differ for a cone instead of a cylinder?
4. What if the cylinder's radius was increasing instead of the height?
5. How would the answer change if the initial radius and height values were different?

**Tip:** In problems involving related rates, always check if any quantity is constant, as it simplifies differentiation.

A potter forms a piece of clay into a right circular cylinder. As she rolls it, the height h of the cylinder increases and the radius r decreases. Assume that no clay is lost in the process. Suppose the height of the cylinder is increasing by 0.4 centimeters per second. What is the rate at which the radius is changing when the radius is 7 centimeters and the height is 10 centimeters?

Explore how to find the rate at which the radius of a cylinder changes when given the rate of height increase. This problem involves related rates and differentiation, focusing on calculus concepts suitable for Grades 11-12. Learn the step-by-step solution, including key formulas like the volume of a cylinder.

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