The Fencing Problem
A farmer has brought 1000 metres of fencing. With this fencing he wants to enclose an area of land. The farmer wants the fencing to enclose an area of the largest size. I will investigate different shapes the fencing can make to achieve the largest area.
Firstly I am going to investigate:
Squares
I am investigate the use of a square with a maximum area and a 1000m perimeter.
The general formula to work out the area for this square is:
AREA= xy
NOT TO SCALE
As the square has four equal sides there can only be one length of each side and one overall area. The length of this side must be:
000m ÷ 4 = 250m
NOT TO SCALE
Therefore the maximum area of the square is 62500m². As I know this, I do not need to display my results in a table or graph as there is only one possible result of a 250m length.
RECTANGLES
To begin with I am going to investigate numerous rectangles that all have a perimeter of 1000 meters, to find out if there is any relevant pattern in my results.
Firstly, the basic formula for quadrilateral shapes such as the rectangle and square is
AREA = LENGTH x WIDTH
In this case it is 498m x 2m = 994m²
NOT TO SCALE
The generalized formula to find out the area of this rectangle is:
2x+2y =1000
x +y=500
Area =xy
=x (500-x)
=500x -x²
NOT TO SCALE
Below is my table of figures for rectangles length, width and area:
Length (m)
x Width
Area (m2)
0
500
0
0
490
4900
20
480
9600
30
470
4100
40
460
8400
50
450
22500
00
400
40000
50
350
52500
200
300
60000
A farmer has brought 1000 metres of fencing. With this fencing he wants to enclose an area of land. The farmer wants the fencing to enclose an area of the largest size. I will investigate different shapes the fencing can make to achieve the largest area.
Firstly I am going to investigate:
Squares
I am investigate the use of a square with a maximum area and a 1000m perimeter.
The general formula to work out the area for this square is:
AREA= xy
NOT TO SCALE
As the square has four equal sides there can only be one length of each side and one overall area. The length of this side must be:
000m ÷ 4 = 250m
NOT TO SCALE
Therefore the maximum area of the square is 62500m². As I know this, I do not need to display my results in a table or graph as there is only one possible result of a 250m length.
RECTANGLES
To begin with I am going to investigate numerous rectangles that all have a perimeter of 1000 meters, to find out if there is any relevant pattern in my results.
Firstly, the basic formula for quadrilateral shapes such as the rectangle and square is
AREA = LENGTH x WIDTH
In this case it is 498m x 2m = 994m²
NOT TO SCALE
The generalized formula to find out the area of this rectangle is:
2x+2y =1000
x +y=500
Area =xy
=x (500-x)
=500x -x²
NOT TO SCALE
Below is my table of figures for rectangles length, width and area:
Length (m)
x Width
Area (m2)
0
500
0
0
490
4900
20
480
9600
30
470
4100
40
460
8400
50
450
22500
00
400
40000
50
350
52500
200
300
60000