Question
ora 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
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
Answers
GPT-4o mini
To find the maximum area that the dog run can have, we need to analyze the quadratic equation given:
\[ A = -w^2 + 60w \]
This equation is in the standard form of a quadratic function \( A(w) = ax^2 + bx + c \), where \( a = -1 \) and \( b = 60 \). The graph of this function will be a downward-opening parabola because \( a \) is negative. The maximum area will occur at the vertex of the parabola.
The formula for the \( w \)-coordinate of the vertex of a quadratic equation \( A(w) = aw^2 + bw + c \) is given by:
\[ w = -\frac{b}{2a} \]
Substituting the values of \( a \) and \( b \):
\[ w = -\frac{60}{2(-1)} = \frac{60}{2} = 30 \]
Now, we can substitute \( w = 30 \) back into the equation for \( A \) to find the maximum area:
\[ A = -30^2 + 60(30) \]
\[ A = -900 + 1800 \]
\[ A = 900 \]
Therefore, the maximum area that the dog run can have is:
**900 square feet.**
\[ A = -w^2 + 60w \]
This equation is in the standard form of a quadratic function \( A(w) = ax^2 + bx + c \), where \( a = -1 \) and \( b = 60 \). The graph of this function will be a downward-opening parabola because \( a \) is negative. The maximum area will occur at the vertex of the parabola.
The formula for the \( w \)-coordinate of the vertex of a quadratic equation \( A(w) = aw^2 + bw + c \) is given by:
\[ w = -\frac{b}{2a} \]
Substituting the values of \( a \) and \( b \):
\[ w = -\frac{60}{2(-1)} = \frac{60}{2} = 30 \]
Now, we can substitute \( w = 30 \) back into the equation for \( A \) to find the maximum area:
\[ A = -30^2 + 60(30) \]
\[ A = -900 + 1800 \]
\[ A = 900 \]
Therefore, the maximum area that the dog run can have is:
**900 square feet.**