A rancher wants to fence in a rectangular area of 23000 square feet in a field and then divide the region in half with a fence down the middle parallel to one side. What is the smallest length of fencing that will be required to do this?
If the length and width are x,y, then we want to minimize
f = 3x+2y subject to
xy = 23000
so, y=23000/x, and we want the minimum of
f = 3x+2(23000/x)
df/dx = 3 - 46000/x^2
df/dx =0 at x = 20/3 √345)
f = 40√345
To find the smallest length of fencing required, we first need to determine the dimensions of the rectangular area. Let's assume the length of the rectangle is 'l' and the width is 'w'.
The total area of the rectangle is given as 23000 square feet, so we have the equation:
l * w = 23000
Next, we divide the region in half with a fence down the middle, parallel to one side. This means we are essentially creating two rectangles of equal area.
Since we are dividing the rectangle in half, the area of each rectangle will be 23000/2 = 11500 square feet.
Now, we have two equations:
l * w = 23000
l * w/2 = 11500
To simplify the calculations, we can rewrite the second equation as:
l * w = 23000 * 2
Now we have two equations with the same value for 'l * w':
l * w = 23000
l * w = 46000
Since the value of 'l * w' is the same in both equations, we can equate them:
23000 = 46000
Dividing both sides of the equation by 23000:
1 = 2
This is not correct. The fact that we got an incorrect equation means that there is no solution to the given problem. The dimensions 'l' and 'w' required to fence in a rectangular area of 23000 square feet and then divide it in half cannot be determined.
Therefore, there is no smallest length of fencing that can be determined for this particular scenario.
To find the smallest length of fencing required, we need to determine the dimensions of the rectangular area first.
Let's assume the length of the rectangular area is 'L' and the width is 'W'.
The area of a rectangle is given by the formula: area = length * width.
In this problem, the area of the rectangular region is given as 23000 square feet.
So, we have the equation: 23000 = L * W ...(Equation 1)
Next, we need to divide the region in half with a fence down the middle parallel to one side. This means that the rectangular area will be split into two equal halves.
Since the fence will be placed down the middle, the width of the rectangular region will remain the same for both halves. So, the width will be denoted as 'W' for both halves of the region.
The length, however, will be halved for each of the two halves. So, we can say that the length of each half is 'L/2'.
Now, let's calculate the area of each half of the region.
Area of each half = (L/2) * W
Since we want both halves to have the same area, we can write the equation:
(L/2) * W = (L/2) * W ...(Equation 2)
From Equation 2, we can see that the left-hand side of the equation represents the area of the first half, while the right-hand side represents the area of the second half.
Combining Equations 1 and 2, we have:
23000 = (L/2) * W
To simplify further, divide both sides of the equation by (L/2):
23000 / (L/2) = W
Simplifying, we get:
46000 / L = W ...(Equation 3)
Now, we have expressed the width 'W' in terms of the length 'L'.
To minimize the length of fencing required, we need to find the minimum perimeter of the rectangular region. The perimeter is given by the formula:
Perimeter = 2 * (L + W)
Substituting the value of W from Equation 3 into the perimeter formula, we get:
Perimeter = 2 * (L + 46000 / L)
To minimize the perimeter, we can take the derivative of this expression with respect to 'L' and set it equal to zero.
Let's differentiate the expression:
d(Perimeter) / dL = 2 * (1 - 46000 / L^2)
Setting it equal to zero:
2 * (1 - 46000 / L^2) = 0
Simplifying further:
1 - 46000 / L^2 = 0
Adding 46000 / L^2 to both sides:
1 = 46000 / L^2
To isolate L^2, divide both sides of the equation by 1:
L^2 = 46000
Taking the square root of both sides:
L = √46000
Calculating the square root, we get:
L ≈ 214.8624
Since the dimensions of a fence must be positive, we take the positive value for L.
Now, substituting this value of L back into Equation 3 to find W:
W = 46000 / L
W ≈ 46000 / 214.8624
W ≈ 214.8624
Therefore, the length of the rectangular area is approximately 214.8624 feet, and the width is also approximately 214.8624 feet.
To find the smallest length of fencing required, we calculate the perimeter using the given formula:
Perimeter = 2 * (L + W)
Substituting the values:
Perimeter = 2 * (214.8624 + 214.8624)
Perimeter ≈ 2 * 429.7248
Perimeter ≈ 859.4496
Therefore, the smallest length of fencing required is approximately 859.45 feet.