To find the domain when the maximum area is achieved, we need to understand the relationship between the width \( w \) and the area \( A \) in the equation \( A = -w^2 + 20w \).
First, note that the equation represents a downward-opening parabola, meaning it will have a maximum point (the vertex). The maximum area will occur at the vertex of the parabola.
The vertex \( w \) of a quadratic function in the form \( A = ax^2 + bx + c \) can be found using the formula:
\[ w = -\frac{b}{2a} \]
In this case, \( a = -1 \) and \( b = 20 \):
\[ w = -\frac{20}{2(-1)} = \frac{20}{2} = 10 \]
Thus, the maximum area occurs when the width \( w \) is 10 yards.
Next, we should consider the domain. The width \( w \) must be non-negative, and since the amount of fencing is limited to 40 yards (forming a rectangular garden with perimeter), the width can go up to 20 yards. The formula for the perimeter \( P \) of a rectangle is given by \( P = 2l + 2w \). For a fence that uses all 40 yards, we set \( P = 40 \):
\[ 2l + 2w = 40 \implies l + w = 20 \]
Assuming \( l \) is the length of the garden, we need to keep \( l \) positive as well, which means:
\[ 0 < w < 20 \]
Therefore, the domain for \( w \) when the maximum area is achieved is:
\[ [0, 20] \text{ yards} \]
However, regarding the multiple-choice answers provided, since the maximum area occurs at \( w = 10 \) yards, this would be the most relevant value to select regarding "when the maximum range is achieved".
So, among the options given in the question, the correct answer reflecting the domain when the maximum area is achieved would be:
10 yards.
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