To find the domain when the maximum area (range) is achieved for the given equation \( A = -w^2 + 20w \), we first recognize that this is a quadratic equation.
The equation can be rewritten in standard form as: \[ A = -w^2 + 20w \]
This is a downward-opening parabola, and to find the maximum area, we can use the vertex formula, which occurs at \( w = -\frac{b}{2a} \). In our equation:
- \( a = -1 \)
- \( b = 20 \)
Calculating the vertex: \[ w = -\frac{20}{2(-1)} = \frac{20}{2} = 10 \]
So, the maximum area occurs when \( w = 10 \) yards.
Now, since \( w \) must be a non-negative width and must also fit within the constraint of the total fencing, we must consider the limitations:
- The total perimeter of the area must equal 40 yards.
- This means if \( w \) is the width and we assume a rectangular garden setup, the length \( l \) can be expressed using the formula for perimeter: \[ 2l + 2w = 40 \] \[ l + w = 20 \] \[ l = 20 - w \]
Given this information:
- When \( w = 0 \): \( l = 20 \), thus \( A = 0 \)
- When \( w = 20 \): \( l = 0 \), thus \( A = 0 \)
Therefore, the domain for \( w \) is from \( 0 \) to \( 20 \) yards, inclusive.
Since the question is asking for the domain when the maximum range (area) is achieved specifically, we can conclude:
- The maximum area occurs at \( w = 10 \) yards.
Thus, the correct response from the provided options regarding the width of the fence when the maximum area is achieved is:
10 yards.