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

10 square yards
10 square yards

100 yards
100 yards

5 yards
5 yards

10 yards
10 yards

Answer 1

To determine the domain of the function \( A = -w^2 + 20w \), we first need to understand the meaning of the function in the context of the fencing problem.

1. **Identify what \( w \) represents:**
- \( w \) is the width of the garden. Since Marilee is using fencing to create a rectangular area, both the width and the length must be positive values.

2. **Understanding the constraints:**
- The total perimeter is given by the total fencing available, which is 40 yards. For a rectangular area, the perimeter \( P \) is given by the formula \( P = 2l + 2w \). If we set \( l \) as the length and \( w \) as the width, we have:
\[
2l + 2w = 40
\]
Simplifying this gives:
\[
l + w = 20 \quad \Rightarrow \quad l = 20 - w
\]
- Since both \( l \) and \( w \) must be positive, we find constraints on \( w \):
- \( w > 0 \) (width must be positive)
- \( l = 20 - w > 0 \) implies \( w < 20 \).

3. **Domain conclusion:**
- Therefore, the domain of \( w \) is \( 0 < w < 20 \).

4. **Finding the maximum area:**
- The area \( A = -w^2 + 20w \) is a quadratic function that opens downwards (since the coefficient of \( w^2 \) is negative), which means it has a maximum value at its vertex.
- The vertex \( w \) can be found using the formula for the vertex of a parabola, \( w = -\frac{b}{2a} \):
\[
w = -\frac{20}{2(-1)} = 10.
\]

5. **Calculate the corresponding area at this width:**
- When \( w = 10 \),
\[
A = -10^2 + 20(10) = -100 + 200 = 100 \text{ square yards.}
\]

Thus, at \( w = 10 \), we achieve the maximum area of 100 square yards, while the width itself is constrained to be in the domain \( (0, 20) \).

In the context of the multiple-choice options provided, the most relevant response regarding the domain when the maximum area is achieved would be:

- **10 yards** (which refers to the width \( w \) at the point of maximum area).

So the correct answer is:

**10 yards**.