A landowner wants to fence in a rectangular area of 60000 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?
as usual, divide the fencing equally among lengths and widths. So, I'll go ahead and predict that the solution will be
7500 x 15000
oops. wrong problem.
y = 60000/x
p = 2x+4y = 2x + 4(60000/x)
dp/dx=0 when x=200√3 and y=100√3
p = 800√3
But, as usual, the length is equally divided among lengths and widths.
To find the shortest total length of fence, we need to determine the dimensions of the rectangle. Let's represent the length of the rectangle as L and the width as W.
Given that the area of the rectangle is 60,000 square meters, we have the equation:
L * W = 60,000
Since we want to divide the rectangle into three parts with two parallel fences, we can assume that the length of the rectangle will be divided into three equal parts, and the width will remain the same. Therefore, we have:
L/3 * W = 60,000
Simplifying the equation, we get:
L * W = 180,000
From here, we need to find the dimensions of the rectangle that minimize the total length of fence. To do so, we'll use the fact that the minimum perimeter occurs when the dimensions are equal. Thus, we have:
L = W
Now we can substitute L in the equation:
L * L = 180,000
L^2 = 180,000
Taking the square root of both sides:
L ≈ 424.26
Now, we can find the width W:
W = L ≈ 424.26
Finally, to calculate the total length of fence required, we need to consider the fences on the lengths and the fences on the widths.
Total length of the fences on the lengths = 3 * L ≈ 3 * 424.26 ≈ 1272.78 meters
Total length of the fences on the widths = 2 * W ≈ 2 * 424.26 ≈ 848.52 meters
Adding these two lengths together gives us the shortest total length of fence:
Total length of fence ≈ 1272.78 + 848.52 ≈ 2121.3 meters
Therefore, the shortest total length of fence that the landowner can use is approximately 2121.3 meters.
To find the shortest total length of fence needed, we need to determine the dimensions of the rectangular area first. Let's assume the length of the rectangle as 'L' and the width as 'W'.
Given that the area of the rectangular area is 60000 square meters, we have:
L * W = 60000
Now, we need to divide this rectangular area into three equal parts using two parallel fences. Let's assume the length of the first part as 'x'. So, the width of the rectangle will remain the same (W), and the length of the remaining two parts will be (L - x).
Now, to calculate the total length of the fences required, let's add up the lengths of all the fences.
1) The length of the first fence (x) will be equal to the length of the rectangle (W).
2) The length of the second fence will be equal to the sum of the two remaining lengths (L - x) needed for the other two parts.
Total fence length = x + 2 * (L - x)
To minimize the total fence length, we need to minimize this expression.
Now, let's substitute the value of L from the area equation:
L = 60000 / W
Total fence length = x + 2 * (60000 / W - x)
To minimize this expression, we need to find the value of x that minimizes the total fence length. We can do this by taking the derivative of the expression with respect to x, setting it equal to zero, and solving for x.
After solving the equation, we can substitute the value of x into the expression for the total fence length to find the minimum value. This will give us the shortest total length of fence that the landowner can use.