Question

You need to create a fenced off region of land for cattle to graze. The grazing area must be a total of 500 square feet, surrounded by a fence, and in the shape of a regular polygon. Within this grazing area, the length of the apothem must be 10 feet long.
Part I. Find the total perimeter of the grazing area.

Part II. If the cost of the fence is $7.95 per linear foot, how much will it cost to place a fence around the entire grazing area?

Part III. Suppose this grazing area is of an industrial grazing area for a major industry farm. What is the total cost to build a fence around the entire land of the large scaled farm? (You have to convert from working with perimeter to area.) also its 1/60 of the larger area.

Include all of the necessary calculations in your final answers for Parts I, II, and III.

Answer 1

Given

A regular polygon with area 500 ft² and apothem 10 ft

Cost of fence is $7.95 per ft

Find

Part III The cost of fence around an area scaled to 60 times the size

Solution

You don't want to think too much about this, because if you do, you find the regular polygon has 3.087 sides. The closest approximation, an equilateral triangle, will have an area of 519.6 ft² for an apothem of 10 ft.

For similar shapes of scale factor "s", the larger shape will have an area of s² times that of the smaller one. Here, it appears the area scale factor s² is 60, so the linear scale factor is

... s² = 60

... s = √60 ≈ 7.7460

The perimeter fence of the 500 ft² area is presumed to be 100 ft long (twice the area of the polygon divided by the apothem—found in Part I), so the perimeter fence of the industrial farm is ...

... (100 ft)×7.7460 = 774.60 ft

and the cost to construct it is

... ($7.95/ft)×(774.60 ft) ≈ $6158