Given the equation \( A = -w^2 + 20w \), we need to determine the domain of \( w \) when the maximum area \( A \) is achieved.
The area \( A \) is a quadratic function opening downwards (because of the negative coefficient in front of \( w^2 \)). The maximum area occurs at the vertex of the quadratic equation.
To find the width \( w \) that gives the maximum area, we use the formula for the vertex of a parabola, which occurs at:
\[ w = -\frac{b}{2a} \]
In this equation, \( A = aw^2 + bw + c \) where:
- \( a = -1 \)
- \( b = 20 \)
- \( c = 0 \)
Substituting into the vertex formula gives:
\[ w = -\frac{20}{2 \times -1} = \frac{20}{2} = 10 \]
Thus, the maximum area is achieved when \( w = 10 \) yards.
Now regarding the domain: The width \( w \) must be non-negative and cannot exceed the total amount of fencing used. Based on the given fencing of 40 yards, the length of the fencing includes both the width and length. The relationship can be described as follows for a rectangular area:
The perimeter \( P \) of the fence:
\[ P = 2l + 2w = 40 \]
This simplifies to:
\[ l + w = 20 \]
From this relationship, the maximum width of the fenced area \( w \) can thus be a maximum of 20 yards when the length is 0, which is logically not feasible in a residential context.
Given this, the domain of \( w \) can be expressed as \( 0 \leq w \leq 20 \).
So, we can summarize that when the maximum area is achieved (at \( w = 10 \)), the domain is:
- \( 0 \leq w \leq 20 \)
However, since the question specifically asks for the range of possible values when the maximum is achieved, "20 yards" is the upper limit of the width when the maximum area is achieved.
Thus, the correct response indicating the width at maximum area (considering the available choices in your question) is:
20 yards.
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