We are given that the sum of two numbers is 18. We need to find the greatest possible product of these two numbers.

1. Problem Description

We are given that the sum of two numbers is

1

8. We need to find the greatest possible product of these two numbers.

2. Solution Steps

Let the two numbers be $x$ and $y$. We are given that $x + y = 18$. We want to maximize the product $P = xy$.

From the equation $x + y = 18$, we can write $y = 18 - x$.

Substituting this into the product equation, we have $P = x(18 - x) = 18x - x^2$.

To find the maximum value of $P$, we can complete the square or use calculus.

Completing the square:

$P = -x^2 + 18x = -(x^2 - 18x)$. To complete the square, we need to add and subtract $(18/2)^2 = 9^2 = 81$ inside the parentheses.

$P = -(x^2 - 18x + 81 - 81) = -(x - 9)^2 + 81$.

Since $-(x - 9)^2$ is always non-positive, the maximum value of $P$ occurs when $x = 9$, and the maximum value is $P = 81$.

Using calculus:

$P = 18x - x^2$. To find the maximum, we take the derivative with respect to $x$ and set it to 0:

$\frac{dP}{dx} = 18 - 2x = 0$.

Solving for $x$, we get $2x = 18$, so $x = 9$.

Then, $y = 18 - x = 18 - 9 = 9$.

The product is $P = xy = 9 \cdot 9 = 81$.

Alternatively, since we are looking for two numbers that add up to 18 and maximize the product, we know that the numbers should be as close as possible. If we allow non-integers, the numbers would be 9 and

9. If we must use integers, then 9 and 9 would give the maximum product.

Let us check values close to

9. $x=8$, $y=10$, $P=80$.

$x=7$, $y=11$, $P=77$.

$x=6$, $y=12$, $P=72$.

So the greatest product is 81, when $x=9$ and $y=9$.

3. Final Answer

81