To determine a reasonable scale for the domain of the equation \( A = -w^2 + 20w \), we first need to consider the constraints of the fencing. The total amount of fencing available is 40 yards, which corresponds to a perimeter measurement for a rectangular area. Since the total perimeter \( P \) of a rectangle is \( P = 2l + 2w \), we can simplify this for our case given the fencing constraint:
\[ 40 = 2l + 2w \quad \Rightarrow \quad l + w = 20 \]
This means the width \( w \) can vary from 0 to 20 yards, since if \( w \) is 0, the length \( l \) will be 20, while if \( w \) is 20, the length \( l \) will be 0.
The area \( A \) is modeled with respect to width \( w \). To examine the valid maximum width for our situation, we understand that \( w \) can range between the boundaries of 0 and 20:
- Minimum width (w = 0): Area = 0
- Maximum width (w = 20): Length = 0, hence Area = 0
Thus, the domain of \( w \) is \( [0, 20] \).
Next, we analyze the options given for a suitable scale for the domain:
- 10
- 1
- 5
- 2
Since the domain range is from 0 to 20, a scale that allows us to cover this distance effectively would be advisable.
Reasonable Choices:
- 1: If we use a scale of 1, we can include all integer widths (0 to 20). This provides the most detailed insight into the values.
- 2: A scale of 2 would allow for widths of 0, 2, 4,..., 20, which gives us a moderate set of values but lacks granularity.
- 5: Would limit the widths to 0, 5, 10, 15, 20 – even less detail.
- 10: Only gives us widths of 0, 10, and 20.
Conclusion:
A reasonable scale for the domain would be 1 so that you can accurately assess all values of \( w \) from 0 to 20. This choice will provide you with the most detailed representation of the widths and their corresponding areas.
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