A rancher wishes to fence in a rectangular plot of land and divide it into two corrals with a fence parallel to two of the sides. If she has 300 yd of fencing available, how large an area can the plot have?
This is what I have so far: A=xy 300=2x+3y
yo moma
To find out the maximum area of the rectangular plot, we can use the fact that the perimeter of the rectangle is equal to the sum of the lengths of all four sides.
Let's assume the length of the rectangle is x and the width is y.
We are given that the rancher has 300 yards of fencing available, which means the perimeter of the rectangle is 300 yards:
Perimeter = 2x + 2y = 300
Now, we know that the fence dividing the plot into two corrals is parallel to two of the sides. This implies that one side of the rectangle is divided into two equal lengths, while the other side remains intact.
Let's say the length of one side divided into two equal lengths is k. This means the length of the other side is x - 2k.
We can now write the perimeter equation in terms of k:
Perimeter = 2(x - 2k) + 2k = 300
Simplifying this equation, we get:
2x - 4k + 2k = 300
2x - 2k = 300
x - k = 150
Now, we also know that the area of a rectangle is given by A = length * width, which in this case is k * (x - 2k) = kx - 2k^2.
To find the maximum area, we need to maximize this expression.
Since we have x - k = 150, we can rewrite kx - 2k^2 as (x - 150) * k.
Now, the maximum area occurs when the expression (x - 150) * k is maximized.
To maximize this expression, we need to set k as large as possible while keeping x - 2k ≥ 0.
The largest possible value for k is x/2, as any larger value for k would make x - 2k negative.
Substituting k = x/2 into the expression, we get:
A = (x - 150) * (x/2) = (x^2 - 300x)/2
Now, we can find the maximum area by finding the critical points of this quadratic function.
Taking the derivative of A with respect to x and setting it equal to 0:
dA/dx = (2x - 300)/2 = x - 150 = 0
Solving for x, we get x = 150.
Substituting this value back into the expression for A, we get:
A = (150^2 - 300 * 150)/2
A = 11250/2
A = 5625
Therefore, the maximum area the plot can have is 5625 square yards.
To solve the problem, we can use the given information and set up equations based on the constraints provided.
Let's assume that the length of one side of the rectangular plot is x yards, and the width is y yards.
Since the rancher wants to divide the plot into two corrals with a fence parallel to two of the sides, we need to consider that the fence will be along the length of the plot, dividing it into two equal parts.
The length of the fence required for the two sides parallel to the length (2x) would be the same as the perimeter. Additionally, we'll need one more side of length y to complete the fence around the rectangular plot. Thus, the total fence length required will be 2x + y.
According to the problem, the rancher has 300 yards of fencing available, so we can set up the equation:
2x + y = 300.
Now, we can rearrange the equation to express y in terms of x:
y = 300 - 2x.
To find the area of the rectangular plot (A), we can use the formula A = length × width, which in this case is A = xy.
Substituting the expression for y into the area equation, we have:
A = x(300 - 2x).
To maximize the area, we need to find the maximum value of A by finding the maximum value of x.
To do this, we can take the derivative of A with respect to x, set it equal to zero, and solve for x. This will give us the value of x that maximizes the area.
By taking the derivative and setting it equal to zero, we get:
300 - 4x = 0.
Simplifying the equation, we have:
4x = 300,
x = 75.
Therefore, the length of one side of the rectangular plot is 75 yards.
Substituting x=75 into the equation y = 300 - 2x, we find:
y = 300 - 150 = 150.
So, the width of the rectangular plot is 150 yards.
Finally, we can find the area by multiplying the length and width:
A = 75 × 150 = 11250 square yards.
Therefore, the largest area the plot can have is 11250 square yards.