You have the rare opportunity to establish a forest preserve

You have the rare opportunity to establish a forest preserve that is 1000 km^2 in area and will be surrounded by agricultural and urban/suburban habitats. You think that detrimental \"edge effects\" (i.e., increased predation rates, cowbird parasitism rates, etc.) may extend ~1 km in from any preserve edge. Which one of the following shapes should you choose for your preserve (assuming that they all cost the same to establish and maintain)? Why? Be sure to calculate the perimeter of each shape (this will be a linear distance in km around the outside of the preserve) and the \"proportion of interior area\" (proportion of total area > 1 knew from any edge; this will be a number between 0 and 1) for each shape. Circle Square Rectangle with length four times the width

Solution

The proposed forest reserve of 1000km² will be surrounded by agricultural and urban areas. The detrimental edge effects will supposedly extend 1km inside from any edge of the reserve. Therefore, we will have to choose a shape of the reserve such that the edge effect is minimum i.e the proportion of the interior area is as high as possible, and the perimeter is as low as possible.

Let us now study the shapes - circle, square and rectangle with length 4 times the width.

a) For a circle, area = r², where =3.14, and r is the radius in km.

Thus, for a circular reserve of 1000km²,

r² = 1000
thus, r² = 1000/3.14
= 318.47
r = 17.846km~17.85km.

Interior area can be calculated by reducing the radius by 1km, and using the same formula as above

Thus, interior area = x (17.85-1)²
= 3.14 x (16.85)2
= 3.14 x 283.92
= 891.51km².

Proportion of interior area = 891.51/1000 = 0.89...........(A)

perimeter = 2r = 2 x 3.14 x 17.85 = 112.1km

b) For a squared reserve, area = L², where L is the length of a side.

Thus, L²=1000
L = 31.62km

now, interior area of a squared reserve would be (L-2)² [since for a square, each side would need to be reduced in length by 1km twice]
= (31.62-2)²
= 29.62²
= 877.34km²

Proportion of interior area = 877.34/1000 = 0.877.......(B)

perimeter = 4 x L = 4 x 31.62 = 126.48km

c) Now, for a rectangular reserve, where the length is 4 times the breadth,

= area = L x B, where L is the length and B is the breadth

as the length is 4 times the breadth, area = 3B x B

thus, 4B x B = 1000
5B² = 1000
B² = 200
B =  14.14km

L = 4B = 4 x 14.14 = 56.56km

Now, interior area, after reducing the length and breadth by 2 = (L-2) x (B-2)
= (56.56-2) x (14.14-2)
= 54.56 x 12.14
= 662.36 km²

Proportion of interior area = 662.36/1000 = 0.66........(C)

perimeter = (2 x L)+(2 x B) = (56.56 x 2) + (14.14 x 2) = 113.12 x 28.28 = 3200 km

Thus, from A, B and C, we know that the proportion of interior area is the highest for a circular reserve i.e. 0.89

Also, the perimeter for a circular reserve is the lowest at 112.1km. Thus, the most logical choice would be to build a circular reserve, which will minimize \'edge effects\'.