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.