Nora is creating a dog run for her dog, Mr. Darcey. She has enough fencing material for a fence with a perimeter of 120 feet. This situation is modeled using the equation A=−w2+60w

Question

𝐴
=
−
𝑤
2
+
60
𝑤
where A
𝐴
is the area and w is the width. Using this equation, identify the maximum range or area the fence can have.(1 point)
12 square feet

Answer 1

To find the maximum area that Nora can enclose using the fence, we start with the area equation given:

\[ A = -w^2 + 60w \]

This is a quadratic equation in the standard form:

\[ A = -w^2 + 60w \]

To find the maximum area, we can use the fact that the maximum value of a quadratic equation in the form \(A = ax^2 + bx + c\) occurs at the vertex, which can be found using the formula:

\[ w = -\frac{b}{2a} \]

In this situation, \(a = -1\) and \(b = 60\):

\[ w = -\frac{60}{2 \times -1} = \frac{60}{2} = 30 \]

Next, we substitute \(w = 30\) back into the area equation to find the maximum area:

\[ A = - (30)^2 + 60 \times 30 \]

Calculating this:

\[ A = -900 + 1800 = 900 \]

Therefore, the maximum area \(A\) that Nora can achieve for the dog run is \(900\) square feet.

So, the correct answer is \(900\) square feet.