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%0D%0A%0D%0A=%0D%0A−%0D%0A%0D%0A2%0D%0A+%0D%0A60%0D%0A%0D%0A where A%0D%0A%0D%0A is the area and w is the width. Using this equation, identify the maximum range or area the fence can have.
To find the maximum area of the fence, we need to find the vertex of the quadratic equation. The formula to find the x-value of the vertex of a quadratic equation in the form A = ax^2 + bx + c is x = -b/(2a).
In this case, the equation is A = -w^2 + 60w.
Comparing this to the form A = ax^2 + bx + c, we have:
a = -1
b = 60
Using the formula x = -b/(2a), we can find the x-value of the vertex:
x = -60/(2 * -1)
x = -60/-2
x = 30
Now we can substitute this x-value back into the equation to find the maximum area:
A = -(30)^2 + 60(30)
A = -900 + 1800
A = 900
Therefore, the maximum area the fence can have is 900 square feet.