A rectangular garden is to be fenced off and divided into 3 equal regions by fencing parallel to one side of the garden. 200 feet of fencing is used. find the dimensions of the garden that maximizes the total area enclosed. what is the maximum area?

To maximize the total area enclosed, we need to find the dimensions of the rectangular garden that will give us the largest possible area. Let's assume the garden has dimensions length L and width W.

Since the garden is divided into 3 equal regions by fencing parallel to one side, we can divide the length into 4 equal parts: L/4, L/4, L/4, and L/4. The width of the garden, W, remains the same.

Therefore, the total length of the fencing required is 4 times the length L/4 plus the width W:

4(L/4) + W = L + W

According to the problem, 200 feet of fencing is used, so we can set up the equation:

L + W = 200

Now, using the formula for the area of a rectangle (A = length × width), we want to maximize the area A:

A = (L/4) × W

We can solve the equation L + W = 200 for L:

L = 200 - W

Now we substitute this value of L in the formula for A:

A = ((200 - W)/4) × W

Expanding the equation:

A = (200W - W^2) / 4

To find the maximum area, we can take the derivative of A with respect to W and set it equal to 0:

dA/dW = (200 - 2W) / 4

Setting dA/dW = 0:

(200 - 2W) / 4 = 0

Simplifying the equation:

200 - 2W = 0

2W = 200

W = 100

Substituting this value of W back into the equation L + W = 200:

L + 100 = 200

L = 100

So, the dimensions of the garden that maximize the total area enclosed are Length = 100 feet and Width = 100 feet.

The maximum area (A) can be calculated as:

A = (L/4) × W = (100/4) × 100 = 2500 square feet

Therefore, the maximum area enclosed is 2500 square feet.

To find the dimensions of the garden that maximize the total area enclosed, we need to follow these steps:

1. Define the variables:
- Let x be the width of the rectangular garden.
- Let y be the length of each of the three equal regions.

2. Express the relationships between the variables:
- The total length of fencing used is given as 200 feet, which means the perimeter of the garden is 200 feet.
- Since there are three equal regions, we can divide the width into three equal parts: x/3.

3. Determine the perimeter equation:
- The perimeter of the garden can be calculated as P = 2x + 4y.
- We know the total length of fencing used is 200 feet, so we have the equation: 2x + 4y = 200.

4. Express the area equation:
- The area of the garden can be calculated as A = x * y.

5. Solve the perimeter equation for y:
- Rearrange the equation 2x + 4y = 200 to isolate y: y = (200 - 2x)/4 = 50 - 0.5x.

6. Substitute the expression for y into the area equation:
- A = x * (50 - 0.5x).

7. Rewrite the area equation in standard form:
- A = -0.5x^2 + 50x.

8. Find the x-coordinate of the vertex of the quadratic equation:
- The x-coordinate of the vertex can be found using the formula: x = -b/(2a).
- In this case, a = -0.5 and b = 50, so x = -50/(2*(-0.5)) = -50/(-1) = 50 feet.

9. Calculate the maximum area:
- Substitute the value of x = 50 back into the area equation: A = -0.5 * 50^2 + 50 * 50 = 1250 square feet.

Therefore, the dimensions of the garden that maximize the total area enclosed are a width of 50 feet and a length of 50 feet for each of the three equal regions. The maximum area enclosed by the fencing is 1250 square feet.

area= LxW

fencing:200=2L+2W+2L
200=4L+2W check that.

area= LW=L(100-50L)
darea/dL=(100-50L)+ L(-50)=0
-100L=-100
L=1
W=100-2L=100-2=98