To find the maximum area that Nora can enclose with her fence, we need to analyze the equation given for the area \( A \) as a function of width \( w \):
\[ A(w) = -w^2 + 60w \]
This is a quadratic equation in the standard form \( A(w) = -w^2 + 60w \).
Step 1: Identify the Vertex
The maximum value of a quadratic equation of the form \( ax^2 + bx + c \) occurs at the vertex, which is given by the formula:
\[ w = -\frac{b}{2a} \]
Here, \( a = -1 \) and \( b = 60 \):
\[ w = -\frac{60}{2 \times -1} = \frac{60}{2} = 30 \]
Step 2: Calculate the Maximum Area
Now, substitute \( w = 30 \) back into the area equation to find the maximum area:
\[ A(30) = - (30)^2 + 60 \times 30 \]
Calculating each term:
\[ A(30) = -900 + 1800 = 900 \]
Conclusion
Thus, the maximum area that Nora can enclose with the fencing material is 900 square feet.
Range of Area
Since the area \( A(w) \) opens downwards (as indicated by the negative leading coefficient in the quadratic), the range of the area can be represented as:
- The minimum is \( 0 \) (when \( w = 0 \)).
- The maximum area we found is \( 900 \).
Therefore, the range of the area \( A \) is:
\[ [0, 900] \]
So, the maximum range of area the fence can have is from 0 ft² to 900 ft².