Question

Nora is creating is creating a dog run for her dog Mr dancing she has enough fencing material for a fence with a perimeter of 120 ft the situation is modeled using the equation h or a equals negative w squared plus 60 w where a is the area and w is the width using this equation identify the maximum range of area the fence can have

Answer 1

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**.