Question

Marilee is creating a garden in her yard. She needs to put up a fence to keep the animals out of it. She has 40 yards of fencing she can erect. This situation is modeled using the equation A=−w2+20w
, where A
is the area of the fence and w is the width of the fence. Using the equation, what is domain when the maximum range is achieved?(1 point)
Responses

100 yards
100 yards

10 yards
10 yards

5 yards
5 yards

10 square yards

Answer 1

To find the domain when the maximum area is achieved, we need to analyze the quadratic equation given:

\[ A = -w^2 + 20w \]

This equation represents a parabola that opens downwards (due to the negative coefficient of \(w^2\)). The maximum area can be found at the vertex of the parabola.

The x-coordinate of the vertex can be calculated using the formula:

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

In this equation:
- \(a = -1\) (the coefficient of \(w^2\))
- \(b = 20\) (the coefficient of \(w\))

Substituting in the values:

\[ w = -\frac{20}{2 \cdot -1} = \frac{20}{2} = 10 \]

So the width \(w\) at which the maximum area occurs is 10 yards.

Next, we also need to consider the possible values for \(w\) based on the constraints of the fencing (40 yards). The perimeter of the rectangular area formed by the width \(w\) and the length \(l\) where \(l\) can be derived from the total fencing gives us:

\[ 2l + 2w = 40 \]
\[ l + w = 20 \]
\[ l = 20 - w \]

Since \(w\) (the width) must be greater than 0 and less than the total perimeter divided by 2 (so that both width and length are positive), we can determine the domain for \(w\):

- \(0 < w < 20\)

Thus, the maximum width is 10 yards within that domain.

Since the question asks for the domain when the maximum range is achieved, the correct response would be:

**10 yards**