A gardenervhas 100 meters of fencing to enclose two adjacent rectangular gardens. The gardener wants the enclosed area to be 400 sq meters. What dimensions should the gardener use to obtain this area?

If the dimensions of the whole garden are x and 2y (because there are two adjacent gardens, each x by y),

2x+4y = 100
2xy = 400

or,

x+2y = 50
xy = 200

since x = 50-2y,

(50-2y)y = 200
50y - 2y^2 = 200
y^2-25y+100 = 0
y = 5 or 20
So, x = 40 or 10

The garden is 10 by 40.

Well, if the gardener wants to enclose two adjacent rectangular gardens and have a total area of 400 sq meters, I suggest they think of it as the great garden fence feud!

Now, let's find a solution to put this feud to rest! The total area of the two gardens combined should be 400 sq meters, meaning each garden should have an area of 200 sq meters.

Since we're dealing with rectangular gardens, let's use 'x' to represent the width of one garden, and 'y' to represent the width of the adjacent garden. We'll have two equations to consider:

Equation 1: x * y = 200 (area of first garden, 'x', multiplied by the area of the second garden, 'y', equals 200 sq meters)
Equation 2: 2x + y = 100 (perimeter of the two gardens equals the length of the fencing, which is 100 meters)

Now, we solve this comedic equation-solving circus act!

From equation 2, we can rearrange it to y = 100 - 2x and substitute it into equation 1:

x * (100 - 2x) = 200
100x - 2x^2 = 200
2x^2 - 100x + 200 = 0

Solve this quadratic equation, and it turns out that x ≈ 7.07 meters (rounded to two decimal places). Substituting this back into equation 2:

2(7.07) + y = 100
14.14 + y = 100
y = 85.86 meters (rounded to two decimal places)

So, the gardener should use dimensions of approximately 7.07 meters by 85.86 meters to achieve an enclosed area of 400 sq meters in total.

Now that's a garden fencing solution that even the feuding vegetables can appreciate!

To find the dimensions of the two adjacent rectangular gardens, we can set up an equation based on the given information.

Let's assume the length of one rectangular garden is x meters. Since the two gardens are adjacent, the length of the other rectangular garden will also be x meters.

The combined width of the two gardens can be calculated as follows:

Width of one garden + Width of the other garden = Total width

To enclose the gardens, the total perimeter should be equal to 100 meters. Therefore:

2(length + width) = 100

2(x + width) = 100

Simplifying further:

x + width = 50

Now we can calculate the area of one garden:

Area = length × width
Area = x × width
Area = 400 sq meters (given)

Substituting the value of width from the previous equation:

x × width = 400
x × (50 - x) = 400

Expanding the equation:

50x - x² = 400

Rearranging the equation to form a quadratic equation:

x² - 50x + 400 = 0

We can solve this quadratic equation to find the value of x, which represents the length of one rectangular garden.

To solve this problem, let's consider the dimensions of the rectangular gardens. Since there are two adjacent gardens, we can assume they share a side.

Let's say the width of one garden is x meters. Then the width of the other garden would also be x meters.

To calculate the length of the gardens, we need to account for the shared side. The combined length of both gardens would be (100 - 2x) meters since we subtract the two sides shared by both rectangular gardens from the total fencing length.

Now, we can calculate the area of each rectangular garden:

Area = Length × Width

For the first garden: Area1 = x × (100 - 2x)
For the second garden: Area2 = x × (100 - 2x)

Since the total area of the enclosed gardens should be 400 square meters:

Area1 + Area2 = 400

Substituting the expressions for Area1 and Area2, we get:

x × (100 - 2x) + x × (100 - 2x) = 400

Simplifying the equation:

2x(100 - 2x) = 400

Expanding and rearranging:

200x - 4x^2 = 400

Bringing the equation to standard form:

4x^2 - 200x + 400 = 0

Now, we can solve this quadratic equation for x. To do this, we can use the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / (2a)

Here, a = 4, b = -200, and c = 400.

Plugging in these values:

x = (-(-200) ± √((-200)^2 - 4 * 4 * 400)) / (2 * 4)

Simplifying:

x = (200 ± √(40000 - 6400)) / 8
x = (200 ± √33600) / 8

Calculating the square root:

x = (200 ± 183.42) / 8

Therefore, we have two possible values for x:

x1 = (200 + 183.42) / 8 ≈ 46.68
x2 = (200 - 183.42) / 8 ≈ 2.08

Now, let's substitute the values of x back into the expressions for the length and width of the gardens to find their dimensions.

For x1 ≈ 46.68:
Length1 = 100 - 2x1 = 100 - 2(46.68) ≈ 6.64
Length2 = 100 - 2x1 = 100 - 2(46.68) ≈ 6.64
Width1 = Width2 = x1 ≈ 46.68

For x2 ≈ 2.08:
Length1 = Length2 = 100 - 2x2 = 100 - 2(2.08) ≈ 95.84
Width1 = Width2 = x2 ≈ 2.08

Therefore, the dimensions the gardener can use to obtain an enclosed area of 400 square meters are either two adjacent gardens with dimensions of approximately 46.68m x 46.68m or approximately 95.84m x 2.08m.