Determine the maximum area of a rectangular piece of land that can be enclosed by 1200 m of fencing.
To determine the maximum area of a rectangular piece of land that can be enclosed by 1200 m of fencing, we need to find the dimensions of the rectangle that would maximize the area.
Let's assume the length of the rectangular piece of land is L (in meters) and the width is W (in meters).
The perimeter of a rectangle is given by the formula:
Perimeter = 2L + 2W
In this case, we know that the perimeter is 1200 m, so we can write the equation as:
1200 = 2L + 2W
We can simplify this equation by dividing both sides by 2:
600 = L + W
Now, let's solve this equation for one of the variables in terms of the other. Let's solve for L:
L = 600 - W
To find the area of a rectangle, we use the formula:
Area = Length * Width
Substituting the value of L from above:
Area = (600 - W) * W
To maximize the area, we need to find the value of W that will result in the maximum area. We can do this by finding the derivative of the area with respect to W and setting it equal to zero:
d(Area) / dW = 0
So, let's take the derivative of the area equation:
d(Area) / dW = (600 - W) - 1 * W
Simplifying:
600 - 2W = 0
Now, we can solve this equation for W:
2W = 600
W = 300
Now, we can plug this value of W back into the equation for L:
L = 600 - W
L = 600 - 300
L = 300
So, the maximum area of the rectangular piece of land that can be enclosed by 1200 m of fencing is when the width is 300 m and the length is 300 m.
To find the maximum area, substitute the values of L and W into the area formula:
Area = Length * Width
Area = 300 * 300
Area = 90000 m^2
Therefore, the maximum area is 90000 square meters.
To determine the maximum area of a rectangular piece of land that can be enclosed by 1200 m of fencing, we need to use the concept of optimization.
Let's assume the length of the rectangular land is L and the width is W. The perimeter of a rectangle is given by the formula P = 2L + 2W. In this case, P = 1200 m.
We want to find the maximum area, which is obtained when the rectangle is a square. In a square, the length and width are equal, so L = W.
Using the perimeter equation, we can rewrite it as:
2L + 2W = 1200
Since L = W, we can substitute L for W:
2L + 2L = 1200
4L = 1200
L = 300
So, the length of the rectangle is 300 m, and the width is also 300 m. Therefore, the maximum area of the rectangular piece of land is 300 m * 300 m = 90,000 square meters.
Area=L*W
Perimeter=2L+2W
L= 600-W
area=600W-W^2
darea/dw=600-2W=0 or W=300
then calculate L, and it is 300 (a square).
max area=LW=9e4