ONLINE DISCUSION
LUIS CEPEDA
A farmer has 12 8-ft long straight fence segments that can be attached to one another to form an
enclosure. To the nearest foot, find the maximum area that can be enclosed with the fencing. In your
response, be sure to describe the enclosure’s shape, what formula you used and how you concluded that
was the maximum area.
1) First what I did is I found the different possible figures I could have used
I tried a square first because it seemed like it would be the biggest area.
96/4 = 24 <- This equals one side length
So I used the area of a square formula which is s2 (s = side) and I got
242 = 576 ft2
24-ft
8-ft
2) Next I decided to test for a rectangle
I decided to try a long rectangle which was 8-ft by 40-ft
I had to use the rectangle formula which was bh (b=base h= height) and I got
40-ft * 8-ft = 320 ft2
40-ft
I saw that the area for this shape was lower than the square so it couldn’t be this one.
3) Next I decided to test a regular hexagon
I know I’m going to have to find the apothem for this in order to find the area
I know each side is 16 and to find the apothem I would have to make a triangle
16-ft
To find the angle there I have to divide 360 by 6 and so I get 60
Then I only need half of this triangle to find the length of my apothem
So I draw an altitude and I’m bisecting the angle and it becomes a
perpendicular bisector on the base so I get a right angle therefore a right
triangle.
So one side of the hexagon is 16-ft long and that line was a perpendicular
and it bisected my base so now my base of this triangle is 8 and it also bisected this angle
so now it equals
30o
And this one equals
600
X
8-ft
To find the length of the apothem which is label X I have to use my trig functions
I see I have to use Either Sin, Cos, or Tan
Well I see I can use Tan because of opposite over adjacent.
So Tan(60 = X/8
I just solve for x so I multiply each side by 8
(8)Tan(60 = X
ONLINE DISCUSION
LUIS CEPEDA
13.85 = X
So now that I have the length of my apothem which is 13.85-ft
So now I have my apothem and the side length of one side of my regular
Hexagon
Now I just got to plug this information into my formula for area for regular
Polygons which is A = (Pa)1/2 P = Perimeter a = apothem
So now I need to find the perimeter
Well each side of the regular hexagon is congruent so all I do is multiply
16 x 6 = 96
Which is 96 so now I have all the info to plug into my formula
A = (96 x 13.85)1/2
A = 664.8
So I saw this one had more area than the Square so I thought maybe its because of the sides are
more outward the shape had more area
4) So then I decided to try a Regular dodecagon
So I did the exact same thing to this one as I did to the hexagon
So I drew in a triangle and I divided 360 by 12 to get this angle
This angle equaled 30o
So again I had to use the regular polygon formula to figure out the are of this
dodecagon
I need the apothem again
So I take the triangle out to solve for the apothem
8-ft
8-ft
I’m going to have to draw an altitude on this triangle which will be my apothem in my dodecagon
And again I get a right angle and it divided my angle in half and my base so now my base equals 4-ft and
my angle equals 150
So now
This angle equals
X
o
75
4-ft
So now I know I have to use a trig function again
I see im using Tan again
So Tan(75 = X/4
Solve for X multiply each side by 4
(4)Tan(75 = X/4(4)
(4)Tan(75 = X
ONLINE DISCUSION
LUIS CEPEDA
14.92 = X
So now I have my apothem which is 14.92-ft
Now I need my perimeter to plug it into my formula
To find it all I have to do is multiply the side length of one side by 12
8 x 12 = 96
And I get 96
So now I plug everything into my formula
A = (96 x 14.92) 1/2
A = 716.16-ft
So now I saw that a dodecagon has the biggest area possible with 12 8-ft long fences.