Question
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.
Answers
GPT-4o mini
To find the maximum area modeled by the equation \( A = -w^2 + 60w \), we need to recognize that this is a quadratic function in the form \( A = -w^2 + 60w \), where the coefficient of \( w^2 \) is negative. This indicates that the parabola opens downwards, which means there is a maximum point (vertex).
The formula for the \( w \) coordinate of the vertex of a quadratic equation \( A = aw^2 + bw + c \) is given by:
\[
w = -\frac{b}{2a}
\]
In our case:
- \( a = -1 \)
- \( b = 60 \)
Substituting the values:
\[
w = -\frac{60}{2 \times -1} = \frac{60}{2} = 30
\]
Now that we have determined the width \( w \) that gives the maximum area, we substitute \( w = 30 \) back into the equation for \( A \):
\[
A = -w^2 + 60w
\]
\[
A = -(30)^2 + 60 \times 30
\]
\[
A = -900 + 1800
\]
\[
A = 900
\]
Thus, the maximum area that the fence can have is \( \boxed{900} \).
The formula for the \( w \) coordinate of the vertex of a quadratic equation \( A = aw^2 + bw + c \) is given by:
\[
w = -\frac{b}{2a}
\]
In our case:
- \( a = -1 \)
- \( b = 60 \)
Substituting the values:
\[
w = -\frac{60}{2 \times -1} = \frac{60}{2} = 30
\]
Now that we have determined the width \( w \) that gives the maximum area, we substitute \( w = 30 \) back into the equation for \( A \):
\[
A = -w^2 + 60w
\]
\[
A = -(30)^2 + 60 \times 30
\]
\[
A = -900 + 1800
\]
\[
A = 900
\]
Thus, the maximum area that the fence can have is \( \boxed{900} \).