Question

A wildlife organization wants to fence off an area of a beach. the organization can only fence 1000 yd2. what length (x) and width (y) should the organization use to use the least amount of fencing as possible?

Answer 1

To fence off an area of 1000 yd² with the least amount of fencing, the wildlife organization should aim to create a square, as a square has the smallest perimeter for a given area. The side length of this square would be the square root of 1000 which is about 31.622 yards.

Step-by-step explanation:

The wildlife organization is seeking to fence off a rectangular area of a beach with a fixed area of 1000 yd2. To use the least amount of fencing as possible, they should aim for the rectangle to have equal length and width because a square has the smallest perimeter for a given area. Let's denote the length of the rectangle as x and the width as y. The area A of a rectangle is A = xy, so we have the equation:

xy = 1000

To find the values of x and y that minimize the perimeter P, where P = 2x + 2y, we use the fact that a square will have the smallest perimeter for a given area.

Since the area is fixed at 1000 yd2, we can set x = y which gives us x2 = 1000. Thus, the value of x (and y) for the least amount of fencing is:

x = y = √1000 ≈ 31.622 yards.

For a square with a side length of approximately 31.622 yards, the total perimeter would be 4x which is approximately 126.49 yards, requiring the least amount of fencing.

Answer 2

We assume in this number that the fence is in a rectangular shape such that its perimeter is equal to,
P = 2x + 2y
From the equation of the area,
A = xy = 1000 ; x = 1000/y
Substituting this to the equation,
P = 2(1000/y) + 2
P = 2000/y + 2y
Getting the differential and equating it to zero.
dP = -2000/y²+ 2 = 0
The value of y from the equation is 31.62 yd
The value of x:
x = 1000/31.62 = 31.62 yd
Therefore, the dimension of the area to be fenced is 31.62 yd by 31.62 yd.