- - - - - - - - - - - - - - ' - ' ; - ' ] - ' =- '/ ' \ - \ \ \ - ;[ [ 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

, 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?

Answer 1

The equation provided, $A=-w^2+20w$, is a quadratic equation that represents a downward-opening parabola. To determine the domain when the maximum area (range) is achieved, we first need to find the vertex of the parabola, which represents the maximum point.

For a quadratic equation of the form $A=a{x}^2+bx+c$, the w-coordinate of the vertex can be found using the formula:

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

In this case, $a=-1$ and $b=20$. Plugging these values into the formula gives:

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

So, the width $w$ that will give the maximum area is 10 yards.

Now, we need to determine the domain of the function. Since the problem states that Marilee has 40 yards of fencing, and the total perimeter $P$ of the rectangular area can be given by:

$$P=2w+2l=40$$

This can be rearranged to find the length $l$:

$$l=20-w$$

For the width and length to be non-negative:

1. $w\ge 0$
2. $20-w\ge 0$ or $w\le 20$

So, the domain of the width $w$ is:

$$0\le w\le 20$$

Thus, when the maximum range (maximum area) is achieved at $w=10$, the domain of the width $w$ is between 0 and 20 yards, inclusive:

Domain: $[0,20]$.

Answer 2

just give me the number of yard -;3-5'7-4]4\5-3=5'4-5'3-594-4';3-0594=3=5--';-]89\0['-[]-\-];]-\]]-$$][\text{[}][$$[]-][][\][][

Answer 3

The range of the width $w$ when the maximum area is achieved is 10 yards.