an experimental farm has 3600 ft. of fencing to create a rectangular area that will be subdivided into 4 equal subregions, what is the largest total area that can be enclosed?
depends. Is the region a 2x2 grid, or a 1x4 grid? In the first case, it will be 600x600. In the 2nd case, it will be 900x360.
To find the largest total area that can be enclosed, we need to determine the dimensions of the rectangular area that will maximize the area.
Let's denote the length of the rectangular area as L and the width as W. Since the rectangular area will be subdivided into 4 equal subregions, we can divide both the length and the width into two equal parts. Therefore, we can express L as 2x and W as 2y, where x and y represent the length and width of each subregion.
Now, we can calculate the perimeter of the rectangular area using the given information about the amount of fencing available:
Perimeter = 2L + 2W = 2(2x) + 2(2y) = 4x + 4y
We know that the perimeter of the rectangular area is given as 3600 ft, so:
4x + 4y = 3600
Dividing both sides of the equation by 4, we get:
x + y = 900
To maximize the area, we need to express one variable in terms of the other. Let's solve this equation for x:
x = 900 - y
Now, we can express the area A in terms of x and y:
A = L * W = (2x) * (2y) = 4xy
Substituting the value of x, we get:
A = 4(900 - y)(y)
Expanding the equation, we have:
A = 3600y - 4y^2
To find the maximum area, we can take the derivative of this equation with respect to y and set it equal to zero:
dA/dy = 3600 - 8y = 0
Solving for y, we get:
8y = 3600
y = 450
Now that we have the value of y, we can substitute it back into the equation for x:
x = 900 - 450 = 450
Therefore, the dimensions of the rectangular area that will maximize the area are:
Length (L) = 2x = 2 * 450 = 900 ft
Width (W) = 2y = 2 * 450 = 900 ft
The largest total area that can be enclosed is:
A = Length * Width = 900 ft * 900 ft = 810,000 square feet.