Marilee is creating a garden in her yard. She needs to put up a fence to keep the animals out. She has 40 yards of fencing. This situation is modeled using the equation A = -w^2 + 20w, where A is the area of the fence and w is the width of the fence.

Question

Using the equation, what is the domain when the maximum area is achieved?

A. 10 yards
B. 5 yards
C. 10 square yards
D. 100 yards

Answer 1

To find the domain for the width \( w \) of the fence when the maximum area is achieved, we can analyze the equation provided:

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

This is a quadratic equation in standard form \( A = -w^2 + 20w \), which graphs as a parabola that opens downwards (because the coefficient of \( w^2 \) is negative).

To find the maximum area, we need to determine the vertex of the parabola. The width \( w \) at the vertex can be found using the formula:

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

where \( a = -1 \) and \( b = 20 \).

Substituting in the values gives us:

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

This indicates that the maximum area occurs at a width of \( w = 10 \).

Next, we consider the constraints for \( w \). Since length cannot be negative, the width must be at least 0 yards. Therefore, the domain for \( w \) (the width of the fence) must be:

\[ 0 \leq w \leq 20 \]

However, the width specified for maximum area is \( w = 10 \). Since the question asks for the domain when the maximum area is achieved, we focus on that specific value.

The correct answer is:

A. 10 yards