Marilee is creating a garden in her yard. She needs to put up a fence to keep the animals out of it. She has 40 yards of fencing she can erect. This situation is modeled using the equation A=−w2+20w
, where A
is the area of the fence and w is the width of the fence. If you were to set up a table to determine the measurements of the width and its corresponding area, what would be a reasonable scale for the domain?(1 point)
Responses
5
5
2
2
1
1
10
10
Rem
When determining a reasonable scale for your table to model the relationship between the width (w) of the garden fence and its corresponding area (A), represented by the equation A = -w² + 20w, you would typically look for a scale that provides you with enough detail without being overwhelming.
The equation is a quadratic one and describes a parabola opening downwards because of the negative coefficient of w². This means the area (A) will increase to a certain point as the width (w) increases, and after reaching its maximum value, the area will start to decrease if the width (w) is increased further.
Considering Marilee has 40 yards of fencing to use and assuming she wants to fence a rectangular area with four sides, the maximum width she could have is half of the total fencing if she wants to make a square, because then all four sides are equal. In that case, w would be 10 yards (because 4 * 10 = 40).
The scale you choose should allow you to understand the behavior of the function in sufficient detail around this point of interest – the maximum area – which would occur before the width reaches 10 yards.
Given this, a scale of:
1. **1**: This scale will give you a very detailed output, allowing you to see the change in area for every yard of width.
2. **2**: This scale is less detailed but still provides a good overview of how the area changes without creating an unnecessarily long table.
3. **5**: This is quite a coarse scale and won't provide as much detail around the critical points, like where the maximum area occurs.
4. **10**: This scale would not be advisable because it would only give you four data points (0, 10, 20, 30) before reaching the maximum possible width, not allowing you to see the change in the area around the maximum.
To strike a balance between detail and manageability, a scale of **2** may be reasonable because it would provide an adequate amount of data points to accurately observe the trend and critical values of the function without resulting in an overly extensive table.
Therefore, the response for a reasonable scale for the domain (width) would be:
**2**