To find the domain of the function \( A = -w^2 + 20w \), we first need to understand the parameters involved.
The variable \( w \) represents the width of the garden. In the context of this problem, the total amount of fencing available is 40 yards, which would be used to create the perimeter of the rectangular garden. The perimeter \( P \) of a rectangle is given by the formula:
\[ P = 2l + 2w \]
Since we have a maximum of 40 yards of fencing, we can simplify this for our width \( w \):
\[ 2l + 2w = 40 \ l + w = 20 \ l = 20 - w \]
Here \( l \) represents the length of the garden. Given that \( w \) must be a non-negative value for the garden to have physical meaning, we have:
\[ 0 \leq w \leq 20 \]
This means that the domain of the function for \( A \) (the area) is:
\[ [0, 20] \]
In this case, as the maximum area is found when the width \( w \) is at its optimal value, which is found by taking the derivative and setting it to zero or completing the square, it can be shown that \( A \) achieves its maximum when \( w = 10 \). The range of area \( A \) can be calculated at the endpoints of the domain.
Now, if the question is about the value when the maximum area (range) is achieved, the maximum occurs at width \( w = 10 \) yards.
Therefore the correct response is:
10 yards