A farmer with 2000 meters 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?
This is a quadratic equation where perimeter is only 3 sides so p=l x 2w and if width is x then side is 2000-2x then area = x(2000-2x) = 2000x-2x^2 the vertex of this parabola will be x=500 and length is 1000 for area=500000
calculus derivative =2000-4x=0 for max-min point and x=500
To find the largest area that can be enclosed, we need to determine the dimensions of the rectangle that can be formed with the given amount of fencing.
Let's denote the width of the rectangle as 'w' and the length as 'l'.
According to the problem, the farmer has 2000 meters of fencing available. We know that the total length of the fencing required is equal to the sum of the widths of the rectangle, two lengths, and the length along the highway. Mathematically, we can represent this as:
2000 = w + 2l + l
Simplifying the equation, we have:
2000 = w + 3l
Now, let's express one variable in terms of the other. Rearranging the equation, we get:
w = 2000 - 3l
The area of the rectangle is given by the product of the width and length:
Area = w * l
Substituting the expression for 'w' in terms of 'l', we have:
Area = (2000 - 3l) * l
Next, we need to find the value of 'l' that maximizes the area. To do this, we can take the derivative of the area function with respect to 'l' and set it equal to zero.
d(Area)/d(l) = 0
Simplifying the equation, we get:
2000 - 6l = 0
Solving for 'l', we find:
6l = 2000
l = 2000 / 6
l = 333.33
Since the length must be a whole number, we can round down to the nearest integer. Therefore, l = 333 meters.
Now, we can substitute the value of 'l' back into the expression for 'w':
w = 2000 - 3l
w = 2000 - 3(333)
w = 2000 - 999
w = 1001
So, the dimensions of the rectangular plot that will maximize the area are 1001 meters (width) and 333 meters (length).
The largest area that can be enclosed is:
Area = w * l
Area = 1001 * 333
Area = 333,333 square meters.
To find the largest area that can be enclosed with a given amount of fencing, we need to determine the dimensions of the rectangular plot that maximize the area.
Let's assume the length of the rectangular plot is L meters and the width is W meters.
We know that the farmer has 2000 meters of fencing, and three sides of the plot will be enclosed with fencing (two sides parallel to the highway, plus one perpendicular side). Therefore, the total length of the three sides will be L + 2W.
According to the problem, one side of the plot is the straight highway, so we don't need any fencing for that side.
We can express this as an equation:
L + 2W = 2000
To solve for either L or W, we need another equation that relates the dimensions of the plot to its area (A).
The area of a rectangle is given by the formula:
A = L * W
Now, we substitute the value of L from the first equation into the area equation:
A = (2000 - 2W) * W
Simplifying:
A = 2000W - 2W^2
To find the largest possible area, we need to find the maximum value of this quadratic equation.
To find the maximum value of a quadratic equation of the form ax^2 + bx + c, we can use calculus by taking the derivative with respect to W and setting it equal to zero.
Let's differentiate the equation with respect to W:
dA/dW = 2000 - 4W
Setting dA/dW = 0:
2000 - 4W = 0
4W = 2000
W = 500
Now, substitute this value of W back into the first equation to find L:
L + 2W = 2000
L + 2(500) = 2000
L + 1000 = 2000
L = 1000
Therefore, the width (W) of the rectangular plot that maximizes the area is 500 meters, and the length (L) is 1000 meters.
Finally, substitute these values into the area equation to find the largest possible area (A):
A = L * W
A = 1000 * 500
A = 500,000 square meters
So, the largest area that can be enclosed is 500,000 square meters.