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 use. This situation is modeled using the equation A=−w2+30w

Question

, where A is the area of the fenced in garden and w is the width of the fence. Using the equation, what is the domain when the maximum range is achieved?(1 point)
Responses

10 yards
10 yards

15 yards
15 yards

100 yards
100 yards

250 yards
250 yards

Answer 1

To find the domain when the maximum area is achieved in the equation $A=-w^2+30w$, let’s first rewrite the equation in terms of the width $w$. The problem also mentions that Marilee has 40 yards of fencing, which means the perimeter of the rectangular garden can be expressed as:

$$P=2l+2w=40$$ This can be simplified to: $$l+w=20\text{(1)}$$ where $l$ is the length of the garden.

Now, substituting for $l$ in terms of $w$ in the area equation gives us: $$A=l\cdot w=(20-w)\cdot w=20w-w^2$$ This is a quadratic function and has the form $A=-w^2+20w$.

The maximum area occurs at the vertex of the parabola defined by this quadratic equation. The width at which the area is maximized can be found using the formula for the vertex $w=-\frac{b}{2a}$:

• Here, $a=-1$ and $b=20$.
• Thus, $$w=-\frac{20}{2\cdot -1}=10$$

This tells us that the maximum area occurs when $w=10$ yards.

Now, let's determine the appropriate domain for the width of the garden. The width, $w$, must be non-negative (since you can’t have a negative width), and it must also be less than or equal to the total width available (which at maximum allows for a total width plus length no greater than 20, as already found in the equation for perimeter).

Therefore, the usable values for $w$ are within the range:

$$0<w<20$$

Consequently, when asking for the domain at which the maximum range is achieved, the correct answer is:

10 yards (the value of $w$ at which the maximum area occurs).