A landowner wants to fence in a rectangular area of 75000 square metres and divide it into three parts with two parallel fences both parallel to one side of the rectangle. Each part will have a different grazing crop for the herds. What is the shortest total length of fence that he can use in metres?
fence length = x = 4w + 2L
L w = 75000 so L = 75000/w
x = 4 w + 150,000/w
dx/dw = 0 for max or min x
0 = 4 -150,000/w^2
w^2 = 150,000/4
w^2 = 37500
w = 193.6
x = 4 w + 150,000/w
x = 4(193.6) + 150,000/193.6
x = 1549 meters
To find the shortest total length of fence required, we first need to determine the dimensions of the rectangular area.
Let's assume the length of the rectangle is L meters and the width is W meters. Since the area of the rectangle is given as 75000 square meters, we have:
L * W = 75000
To divide the rectangle into three equal parts, we will have two parallel fences dividing it into three sections.
Without loss of generality, let's consider the length of the rectangle as the side parallel to the fences. So, we will divide the length (L) into three equal portions: L/3, L/3, and L/3.
Now, let's calculate the perimeter of each section:
Perimeter of the first section = (length + width + length) = (L/3 + W + L/3) = (2L/3 + W)
Perimeter of the second section = (length + width + length) = (L/3 + W + L/3) = (2L/3 + W)
Perimeter of the third section = (length + width) = (L/3 + W)
To find the total length of the fence, we need to add the perimeters of all three sections:
Total length of fence = Perimeter of the first section + Perimeter of the second section + Perimeter of the third section
= (2L/3 + W) + (2L/3 + W) + (L/3 + W)
= (4L/3 + 2W) + (L/3 + W)
= (5L/3 + 3W)
Our goal is to minimize this total length of fence, so we need to find the minimum value of (5L/3 + 3W).
Since we know the area of the rectangle is 75000 square meters, we can substitute W = 75000/L in the perimeter equation:
Total length of fence = (5L/3 + 3W)
= (5L/3 + 3 * 75000/L)
To simplify the equation, let's multiply both sides by 3 to get rid of the fractions:
3 * Total length of fence = 5L + 225000/L
Now, the problem is reduced to finding the minimum value of 5L + 225000/L.
To find the minimum, we can take the derivative of the expression with respect to L and set it equal to zero:
d(5L + 225000/L)/dL = 5 - 225000/L^2 = 0
Solving this equation will give us L, which we can then substitute back into the equation to find the minimum value of the total length of fence.