Quadratic Functions from Situations Practice

Question

Complete this assessment to review what you’ve learned. It will not count toward your grade.
2 of 52 of 5 Items

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
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)
square feet
Remaining Attempts : 3

Answer 1

To find the maximum area that can be enclosed by the fence, we need to analyze the quadratic equation given for the area:

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

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

The maximum value of a quadratic function occurs at the vertex, which can be found using the formula:

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

Substituting \( a \) and \( b \):

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

Now that we have the width \( w \), we can substitute it back into the equation to find the maximum area \( A \):

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

Thus, the maximum area that the fence can have is:

900 square feet.