A rancher wants to fence in an area of 1500000 square feet in a rectangular field and then divide it in half with a fence down the middle, parallel to one side.
What is the shortest length of fence that the rancher can use?
let each of the 2 equal sides by x
let yeach of the 3 equal sides by y
xy = 1,500,000
y = 1500000/x
let the length be L
L = 2x + 3y
= 2x + 4500000/x
dL/dx = 2 - 4500000/x^2
= 0 for a min of L
2x^2 = 4500000
x^2 = 2250000
x = 1500
then y = 1500000/1500 =1000
so L = 2(1500) + 3(1000) = 6000 ft
Oh, sorry, I thought you meant just the length of that splitting fence.
To find the shortest length of fence that the rancher can use, we first need to determine the dimensions of the rectangular field.
Let's assume the rectangular field has length "L" and width "W". Therefore, the area of the field can be expressed as:
Area = Length × Width
From the given information, we know that the area of the field is 1,500,000 square feet. So, we have the equation:
1,500,000 = L × W
Now, the rancher wants to divide the field into two equal halves with a fence down the middle. This means that the length of the fence will be equal to the width of the field.
We need to minimize the amount of fence needed, so we should make the field as square as possible. In other words, we need to find the dimensions of the field (L and W) that will make it as close to a square as possible.
To find the dimensions that make the field as square as possible, we can start by finding the square root of the area (1,500,000). This will give us the length and width of the field if it were a perfect square.
√1,500,000 ≈ 1224.74
Now, we should check if this number is a perfect square. If it is, we have found the dimensions of the field (L and W), and we can proceed with finding the length of the fence.
However, if 1224.74 is not a perfect square, we can try the next closest integers. In this case, the closest integers are 1225 and 1224.
Let's check if 1225 is a perfect square:
√1225 = 35
Since 1225 is a perfect square, we have found the dimensions of the field. We can conclude that the field has a length of 35 feet and a width of 35 feet.
Now, we can calculate the length of the fence.
The fence down the middle of the field will have a length equal to the width of the field, which is 35 feet. However, since we need to divide the field in half, we only need half of the length, which is (35 / 2) = 17.5 feet.
Finally, we need to add the length of the fence that goes around the rectangular field. Since there are two lengths and two widths of the field, the remaining part of the fence has a length of 2L + 2W.
Substituting the values, we get:
2(35) + 2(35) = 70 + 70 = 140 feet
Therefore, the shortest length of fence that the rancher can use is 140 feet.
.000000000000001 feet or perhaps the length of the width of a cow. :)
The problem does not make much sense as stated. You could make the field extremely longs and not at all wide and divide it with a little fence in the middle the length of your very short ends.