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

where A
is the area and w is the width. Using this equation, identify the maximum range or area the fence can have.(1 point)
its not 30 square feet

Answer 1

To find the maximum area of the dog run, we can analyze the given equation for area, which is:

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

This is a quadratic equation in the standard form \( A = ax^2 + bx + c \), where \( a = -1 \), \( b = 60 \), and \( c = 0 \).

The maximum value of a quadratic equation in this form occurs at the vertex, which can be found using the formula:

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

Substituting in the values for \( a \) and \( b \):

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

Now that we have the width \( w = 30 \) feet, we can find the corresponding area by substituting this value back into the area equation:

\[ A = -30^2 + 60(30) \] \[ A = -900 + 1800 \] \[ A = 900 \]

Thus, the maximum area of the fence that can be created is:

900 square feet.