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**Answer:**

6 m

**Step-by-step explanation:**

Let p represent the length of the perimeter fence. Let x represent the length parallel to the house. Then the width perpendicular to the house is ...

w = (p -x)/2

and the area of the garden is ...

area = length · width

area = x(p -x)/2

For a fixed perimeter length p, this equation describes a parabola that opens downward. Its vertex is on the line of symmetry, halfway between the zeros. The area will be zero when x=0 and when x=p, so the vertex (maximum area) is located at ...

x = (0 +p)/2 . . . . . average of zero values

x = p/2

Here, we have p=12 m, so p/2 = 6 m, and the width is (12 -6)/2 = 3 m.

**The length of the maximum-area garden is 6 meters**.

_____

Its area is 18 square meters.

**Answer:**

- 6 m

**Step-by-step explanation:**

Let the width be w and the length be l

__We have the measure of 3 sides:__

- 2w + l = 12

__Area of the garden is:__

- A = lw

__Substitute w with l using the first equation:__

- A = lw =
- l(12 - l)/2 =
- -1/2( l² - 12l) =
- -1/2(l² - 12l + 36 - 36) =
- -1/2(l - 6)² + 18

The final expression for the area can have maximum value when l - 6 = 0 because the first part of expression gets negative value for any l ≠ 6

And the maximum possible area is 18 m²

**So l = 6 m is the answer for maximum area**

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