The sum of two numbers is 18. The greatest product of these two numbers can be :

Options

A. 17
B. 81
C. 80
D. can't be determined

Answer

To find the greatest product of two numbers whose sum is 18, we can follow these steps:

### Step-by-Step Solution:

1. **Define the Variables**: Let the two numbers be \( x \) and \( y \).

\[ x + y = 18 \]

2. **Express One Variable in Terms of the Other**: From the equation above, we can express \( y \) in terms of \( x \):

\[ y = 18 - x \]

3. **Write the Product Function**: The product \( P \) of the two numbers can be expressed as:

\[ P = x \cdot y = x(18 - x) = 18x - x^2 \]

4. **Find the Maximum Product**: To find the maximum product, we need to find the vertex of the quadratic equation \( P = -x^2 + 18x \). The vertex form of a quadratic equation \( ax^2 + bx + c \) gives the maximum value at \( x = -\frac{b}{2a} \).

Here, \( a = -1 \) and \( b = 18 \):

\[ x = -\frac{18}{2 \cdot -1} = \frac{18}{2} = 9 \]

5. **Calculate the Corresponding Value of \( y \)**: Substitute \( x = 9 \) back into the equation for \( y \):

\[ y = 18 - 9 = 9 \]

6. **Calculate the Maximum Product**: Now, calculate the product:

\[ P = x \cdot y = 9 \cdot 9 = 81 \]

### Conclusion:
The greatest product of the two numbers whose sum is 18 is **81**.

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