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

Question

A
=
−
w
2
+
20
w
, where A
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

5 yards
5 yards

10 yards
10 yards

100 yards

Answer 1

The equation given for the area $A$ of the garden in terms of the width $w$ is:

$$A=-w^2+20w$$

This is a quadratic equation that opens downward, indicating that it has a maximum value. To find the width $w$ at which the maximum area occurs, we can use the vertex formula for a parabola, given by:

$$w=-\frac{b}{2a}$$

In this equation, $a=-1$ and $b=20$. Plugging in these values:

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

Thus, the width at which the maximum area is achieved is $w=10$ yards.

Next, we need to find the corresponding area $A$ when $w=10$:

$$A=-({10}^2)+20(10)=-100+200=100\text{ square yards}$$

So, the maximum area is $100$ square yards, which is also the maximum range of the function $A(w)$.

Regarding the question about the domain when the maximum range is achieved: the width $w$ must be in the range that is valid for the fencing. Since the total fencing is 40 yards, and the relationship in the fencing situation is $2w+2l=40$, where $l$ is the length, we can state that the dimensions must be positive.

However, in the context of the provided answers and the context of asking for the "domain when the maximum range is achieved", it looks like they might be mixing terms.

To succinctly answer the original multiple-choice question regarding when the maximum area is achieved:

• The domain of the widths obtainable would be $0<w<20$ (since if $w$ were to exceed 20, $l$ would not remain a positive number).
• The area at the maximum is given as $100$ square yards.

Among the multiple-choice answers, the correct one (the area when the maximum range is achieved) is:

100 square yards

The closest option relevant for domain in terms of the width is not explicitly listed. However, if we're strictly talking about the maximum area being achievable, that answer is indeed not present directly in the options provided.

If using specific choices:

• The main relevant result seems closest to 100 but if considering only about the choices provided (which may potentially correlate with $w$), the critical number would still be 10 yards (the width that gives the maximum area).

So in summary, the relevant interpretations from this would revolve primarily around 10 yards in terms of specifications for width.