A rancher has
4 comma 7004,700
feet of fencing available to enclose a rectangular area bordering a river. He wants to separate his cows and horses by dividing the enclosure into two equal areas. If no fencing is required along the river, find the length of the center partition that will yield the maximum area.
A rancher has
4,700 feet of fencing available to enclose a rectangular area bordering a river. He wants to separate his cows and horses by dividing the enclosure into two equal areas. If no fencing is required along the river, find the length of the center partition that will yield the maximum area.
You don't say whether the dividing fence is to be parallel to the river or not. If not, then let x be the length of the side parallel to the river. Then you have
x+3y = 4700
so, x = 4700-3y
The area is
a = xy = (4700-3y)y = 4700y - 3y^2
The vertex of this parabola is at (2350/3,5522500/3) or (783.33,1.84*10^6)
As usual, the maximum area is when the fence is divided equally among lengths and widths:
one length of 2350
3 widths of 2350/3
Now, if the dividing fence op length x is parallel to the river, you have
2x+2y = 4700
a = x(2350-x) = 2350x-x^2
and max area is at x=2350/2 and is 1.38*10^6
To find the length of the center partition that will yield the maximum area, we can follow these steps:
1. Let's assume that the length of the entire rectangular enclosure is L.
2. Since the enclosure is divided into two equal areas, the length of each half is L/2.
3. The width of the enclosure can be represented as W.
4. The perimeter of the enclosure is given as 4,700 feet, which includes the length of the entire enclosure and the two widths.
5. Using the perimeter, we can set up an equation: 2L + 2W = 4,700.
6. Since no fencing is required along the river, the width of the enclosure will be the same as the length of the center partition that separates the cows and horses.
7. Let's substitute W with L/2 in the perimeter equation: 2L + 2(L/2) = 4,700.
8. Simplifying the equation, we get: 2L + L = 4,700.
9. Combining like terms, we find: 3L = 4,700.
10. Dividing both sides by 3, we get: L ≈ 1,566.67.
Therefore, the length of the center partition that yields the maximum area is approximately 1,566.67 feet.