I've tried this many times, but I keep getting a really big answer.
A rancher wants to fence in an area of 2500000 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?
A = a b so b = A/a
P = 2 a + 3 b
P = 2 a + 3 A/a
now A=2.5*10^6
P = 2a + 7.5*10^6/a
dP/da = 0 at in or max
dP/da = 0 = 2 -7.5*10^6/a^2
a^2 = 3.75*10^6
a = 1.936*10^3
then b = 2.5*10^6/1.936*10^3
b = 1.291*10^3
P = 2*a+3*b = 7.745*10^3
To find the shortest length of fence that the rancher can use, we need to determine the dimensions of the rectangular field first.
Let's begin by breaking down the problem step by step:
1. Start by finding the square root of the given area of 2,500,000 square feet.
√2500000 ≈ 1581.14
2. Now, we have the length of one side of the rectangle, which is approximately 1581.14 feet.
3. Since the rancher wants to divide the field into two equal halves with a fence down the middle, we need to find the width of the rectangle.
4. Divide the area by the length we just found:
Area ÷ Length = Width
2500000 ÷ 1581.14 ≈ 1581.14
5. The width of the rectangle is approximately 1581.14 feet.
6. Since we are dividing the field in half with a fence, we only need the length of one side and the width to calculate the total length of fence.
7. To find the total length of fence needed, double the length and add the width:
(Length × 2) + Width = Total Length of Fence
(1581.14 × 2) + 1581.14 ≈ 4743.42 feet
Therefore, the shortest length of fence that the rancher can use to enclose and divide the rectangular field is approximately 4743.42 feet.