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The problem provides the area of a garden, \( A \), in square meters, given by:

\[
A = 3x(200 - 3x)
\]

It asks to show that the area cannot exceed \( 10,000 \, \text{m}^2 \).

### Solution:

1. **Expand the equation**: 

   First, expand the given expression for \( A \):

   \[
   A = 3x(200 - 3x) = 600x - 9x^2
   \]

   So, the equation becomes:

   \[
   A = -9x^2 + 600x
   \]

2. **Identify the type of function**:

   This is a quadratic function of the form \( A(x) = -9x^2 + 600x \), which represents a parabola that opens downwards (since the coefficient of \( x^2 \) is negative). The maximum value of \( A \) occurs at the vertex of the parabola.

3. **Find the vertex**:

   The \( x \)-coordinate of the vertex of a parabola \( ax^2 + bx + c \) is given by the formula:

   \[
   x_{\text{vertex}} = -\frac{b}{2a}
   \]

   Here, \( a = -9 \) and \( b = 600 \). Substituting these values:

   \[
   x_{\text{vertex}} = -\frac{600}{2(-9)} = \frac{600}{18} = 33.33
   \]

4. **Calculate the maximum area**:

   Now, substitute \( x = 33.33 \) back into the expression for \( A \) to find the maximum area:

   \[
   A = -9(33.33)^2 + 600(33.33)
   \]

   Let's calculate this:

   \[
   A = -9(1111.11) + 600(33.33) = -9999.99 + 19998 = 9998.01
   \]

   The maximum area is approximately \( 9998 \, \text{m}^2 \), which is very close to 10,000 m² but does not exceed it.

### Conclusion:
The maximum area of the garden is approximately \( 9998 \, \text{m}^2 \), so the area cannot exceed \( 10,000 \, \text{m}^2 \).

Would you like further details or explanations on any step?

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Here are 5 related questions you can consider:
1. How do you find the vertex of any quadratic function?
2. Can the garden have a negative area? Why or why not?
3. How can we verify the maximum value of a quadratic function algebraically?
4. What are the implications of the negative coefficient of \( x^2 \) in real-world problems?
5. How can you graph \( A = -9x^2 + 600x \) to visualize the solution?

**Tip**: When dealing with quadratic functions, always consider if the parabola opens upwards or downwards to determine if it has a maximum or minimum value.

The area of a garden, A m², is given by A = 3x(200 - 3x). Show that the area of the garden cannot exceed 10,000 m².

Learn how to determine the maximum area of a garden modeled by a quadratic function A = 3x(200 - 3x), and prove that it cannot exceed 10,000 m². This problem covers quadratic functions, vertex calculations, and real-world applications of algebraic concepts.

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