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The question asks for the maximum value of the product \( xy \) given that the sum of two positive numbers \( x \) and \( y \) is 18. This can be approached by setting up the following conditions:

1. **Given condition:** \( x + y = 18 \).
2. **Objective:** Maximize \( xy \).

### Solution:
Since \( x + y = 18 \), we can express \( y \) in terms of \( x \):  
\[ y = 18 - x \]

Now, substitute \( y \) in the product \( xy \):
\[ xy = x(18 - x) \]
\[ xy = 18x - x^2 \]

To find the maximum value, we differentiate \( xy \) with respect to \( x \) and set the derivative to zero:
\[ \frac{d(xy)}{dx} = 18 - 2x \]
Setting \( \frac{d(xy)}{dx} = 0 \):
\[ 18 - 2x = 0 \]
\[ x = 9 \]

Substitute \( x = 9 \) back into the equation for \( y \):
\[ y = 18 - 9 = 9 \]

Thus, when \( x = 9 \) and \( y = 9 \), the product \( xy \) is:
\[ xy = 9 \times 9 = 81 \]

### Answer:
The maximum value of \( xy \) is **81**.

Would you like further details or have any questions?

Here are some related questions:
1. What if \( x + y \) was a different number, how would the answer change?
2. How does the constraint \( x + y = 18 \) affect maximizing \( xy \)?
3. Can this problem be solved using another method, like completing the square?
4. How would we approach this if we wanted a minimum value for \( xy \)?
5. What if \( x \) and \( y \) were restricted to integers only?

**Tip:** When dealing with maximum or minimum product problems under a sum constraint, expressing one variable in terms of the other can simplify the optimization process.

Jumlah bilangan positif x dan y adalah 18. Nilai maksimum xy adalah ...

Explore how to find the maximum value of the product xy given that the sum of two positive numbers x and y is 18. This solution uses algebraic techniques and derivatives for optimization, suitable for high school students.

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