A farmer with 3000 feet of fencing wants to enclose a rectangular plot that borders on a straight highway. If the farmer does not fence the side along the highway, what is the largest area that can be enclosed?
Area = A x B
Total length = 3000 = A + 2B
B = (3000-A)/2 so,
Area = A x (3000-A)/2 =
= 3000A/2 - (1/2)A^2
take 1st derivative
0 = 1500 - A and solve for A
A = 1500 and B = 750
To find the largest area that can be enclosed with 3000 feet of fencing, we need to determine the dimensions of the rectangular plot.
Let's assume the length of the rectangular plot is L, and the width is W.
The total amount of fencing required will be the sum of all four sides of the rectangle:
2L + W = 3000 ------------ (1) (since the side along the highway is not fenced)
We need to express the area (A) in terms of a single variable. Since A = L * W, we can solve the equation (1) for L:
2L = 3000 - W
L = (3000 - W) / 2
Substituting this value for L in the area equation:
A = L * W = [(3000 - W) / 2] * W
Now we have the area expressed in terms of a single variable, W. To find the largest possible area, we can take the derivative of A with respect to W and find where it equals zero:
dA/dW = [(3000 - W) / 2] - (W / 2) = 0
Simplifying,
3000 - W - W = 0
3000 - 2W = 0
2W = 3000
W = 1500
Substituting this value of W back into the equation (1) to find L:
2L + 1500 = 3000
2L = 3000 - 1500
2L = 1500
L = 750
Therefore, the largest area that can be enclosed with 3000 feet of fencing is when the length is 750 feet and the width is 1500 feet.
To determine the largest area that can be enclosed by the fencing, we need to find the dimensions of the rectangle that maximize the area. Let's break down the problem into steps:
Step 1: Define the variables:
Let's assume that the width of the rectangle is x (in feet).
Since the rectangle only has three sides fenced (excluding the side along the highway), the length of the rectangle will be (3000 - x) feet.
Step 2: Calculate the perimeter:
The perimeter of a rectangle is calculated by adding up all the sides. In this case, we have:
Perimeter = 2(width + length)
Perimeter = 2(x + (3000 - x))
Perimeter = 2(3000)
Perimeter = 6000 feet
Step 3: Use the perimeter formula to solve for the variable x:
Now, we need to solve for x in the perimeter equation:
6000 = 2(x + (3000 - x))
Simplifying the equation:
6000 = 2(3000 + 0)
6000 = 6000
As you can see, no matter what value we assign to x, the perimeter remains constant at 6000 feet. Therefore, in order to maximize the area, we need to find the dimensions that give us a square, as a square has the largest area for a given perimeter.
Step 4: Derive the area formula:
The area of the rectangle is calculated by multiplying the width by the length:
Area = width * length
Area = x * (3000 - x)
Step 5: Find the maximum area:
To determine the maximum area, we find the vertex of the quadratic equation. We can achieve this by using calculus or by using symmetry properties of the parabolic shape.
The symmetry of the parabola suggests that the maximum area occurs when x is halfway between 0 and 3000. So, the maximum area occurs when x = 1500 feet.
Substituting x = 1500 into the area equation:
Area = 1500 * (3000 - 1500)
Area = 1500 * 1500
Area = 2,250,000 square feet
Therefore, the largest area that can be enclosed by the fencing is 2,250,000 square feet.