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

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
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.

GPT-4o mini · Answer

To find the maximum area that Nora can achieve for her dog run, we start with the equation given:

$A = - w^{2} + 60 w$

This equation represents a quadratic function in the standard form $A = a x^{2} + b x + c$ , where:

$a = - 1$ (which indicates that the parabola opens downwards),
$b = 60$ ,
$c = 0$ .

The maximum value of a quadratic function occurs at the vertex. The formula for the width $w$ at the vertex for a quadratic equation $A = a x^{2} + b x + c$ is given by:

$w = - \frac{b}{2 a}$

Substituting the values of $a$ and $b$ :

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

Now that we have the width $w = 30$ feet, we can substitute this back into the area equation to find the maximum area:

$A = - w^{2} + 60 w$ $A = - (30)^{2} + 60 \times 30$ $A = - 900 + 1800$ $A = 900$

Thus, the maximum area that the dog run can have is 900 square feet.