hard math promblem

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

Answer 1

Looks like no one wanted to answer this one!

We want to specify a regular polygon with an apothem of 10 feet and an area of 500 square feet.

From https://mathmonks.com/polygon/area-of-polygon we know that the area of a regular polygon is given by:

A = (1/2) * p * a

where p = perimeter and a is the apothem. The perimeter can be written as n * s where n = number of sides, each of which is s feet long. From the formula:

A = (1/2) * n * s ft * 10 ft = 500 ft^2

or

n * s = p = 1000 ft^2 / 10 ft = 100 ft

The problem has many solutions. Note that a regular polygon has 3 or more sides.

n s

=======

3 33.33 ft

4 25 ft

5 20 ft

...

10 10 ft

50 2 ft

100 1 ft

etc.