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The problem reads as follows:

Each side of a square is increasing at a rate of \(6 \, \text{cm/s}\). At what rate (in \(\text{cm}^2/\text{s}\)) is the area of the square increasing when the area of the square is \(9 \, \text{cm}^2\)?

### Solution:

#### 1. Relation between area and side of the square:
The area \(A\) of a square is given by:
\[
A = s^2
\]
where \(s\) is the side length of the square.

#### 2. Differentiating with respect to time \(t\):
\[
\frac{dA}{dt} = 2s \frac{ds}{dt}
\]
Here:
- \(\frac{dA}{dt}\) is the rate of change of area (what we need to find),
- \(s\) is the side length of the square,
- \(\frac{ds}{dt}\) is the rate of change of the side length, which is given as \(6 \, \text{cm/s}\).

#### 3. Finding \(s\) when \(A = 9 \, \text{cm}^2\):
If \(A = s^2\), then:
\[
s = \sqrt{A} = \sqrt{9} = 3 \, \text{cm}.
\]

#### 4. Substituting values into the derivative:
\[
\frac{dA}{dt} = 2s \frac{ds}{dt} = 2(3)(6) = 36 \, \text{cm}^2/\text{s}.
\]

### Final Answer:
The area of the square is increasing at a rate of:
\[
\boxed{36 \, \text{cm}^2/\text{s}}
\]

Let me know if you’d like a more detailed explanation or have further questions!  

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### Related Questions:
1. How would the answer change if the area were 16 cm² instead of 9 cm²?
2. What happens to the rate of change of the area as the side length increases further?
3. Can you derive the formula for the rate of change of the area geometrically?
4. What if the side length were decreasing at the same rate instead of increasing?
5. How can this concept be applied to non-square shapes like rectangles?

### Tip:
Always relate the changing quantity (like area) to the given rates (like side length) using derivatives when solving related rates problems!

Each side of a square is increasing at a rate of 6 cm/s. At what rate (in cm²/s) is the area of the square increasing when the area of the square is 9 cm²?

Solve a related rates problem involving a square where each side is increasing at a rate of 6 cm/s. Learn how to find the rate of change of the area of the square when the area is 9 cm². This problem applies the chain rule and related rates concepts in calculus. Suitable for students in Grades 10-12.

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