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?

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.