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
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
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.
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.