There is a need to make a fence that is 1000m long. The area inside the fence has to have the maximum area. I am investigating which shape would give this.
Rectangles:
I am going to start investigating different shape rectangles, all which have a perimeter of 1000m. Below are 2 rectangles (not to scale) showing how different shapes with the same perimeter can have different areas.
Here are some pure examples of what I have to accomplish with rectangles having perimeters of 1000 metres.
E.g. 1:
E.g. 2:
In a rectangle (of 1000m), any two different length sides will add up to 500m, this is because each side has an opposite with the same length. Therefore in a rectangle of 100m X 400m, there are two sides opposite each other that are 100m long and two sides next to them that are opposite each other that are 400m long. This means that you can work out the area if you only have the length of one side. To work out the area of a rectangle with a base length of 200m, I subtract 200m from 500m, giving me 300m and then times 200m by 300m.
I can then put this into an equation form: 1000 = X (500 -X)
X = Base Length
Below is a table of results, worked out using the formula above. I have gone down the table by taking 10m off the base every time.
There are three columns one showing the height (m) another showing the base length and the final column shows the area (m2).
Base Length
Height (m)
Area (m2)
500
0
0
490
0
4900
480
20
9600
470
30
4100
460
40
8400
450
50
22500
440
60
26400
430
70
30100
420
80
33600
410
90
36900
400
00
40000
390
10
42900
380
20
45600
370
30
48100
360
40
50400
350
50
52500
340
60
54400
330
70
56100
320
80
57600
310
90
58900
300
200
60000
290
210
60900
280
220
61600
270
230
62100
260
240
62400
250
250
62500
240
260
62400
230
270
62100
220
280
61600
210
290
60900
200
300
60000
90
310
58900
80
320
57600
70
330
56100
60
340
54400
50
350
52500
40
360
50400
30
370
48100
20
380
45600
10
390
42900
00
400
40000
90
410
36900
80
420
33600
70
430
30100
60
440
26400
50
450
22500
40
460
8400
30
470
4100
20
480
9600
0
490
4900
0
500
0
I can draw a graph of base length against area using this formula, (Fig 1.0).
According to the table and the graph, the rectangle with a base of 250m has the greatest area. As you can see in the table below I have gone into greater depth to find if the rectangle with a base of 250m is the highest possible area. This shape is also called a square, or a regular quadrilateral. Because I only measured to the nearest 10m, I cannot tell whether the graph is true, and does not go up just to the sides of 250m, so I will have to investigate further by focusing around 250m. I will work out the results using 249m, 249.5 and 249.75, as shown below.
Height (m)
Base (m)
Area (m2)
251
249
62499
250.5
249.5
62499.75
250.25
249.75
62499.9375
250
250
62500
249.75
250.25
62499.9375
249.5
250.5
62499.75
249
251
62499
All of these results fit into the graph line that I have, making my graph reliable, Fig 1.0.
Rectangles:
I am going to start investigating different shape rectangles, all which have a perimeter of 1000m. Below are 2 rectangles (not to scale) showing how different shapes with the same perimeter can have different areas.
Here are some pure examples of what I have to accomplish with rectangles having perimeters of 1000 metres.
E.g. 1:
E.g. 2:
In a rectangle (of 1000m), any two different length sides will add up to 500m, this is because each side has an opposite with the same length. Therefore in a rectangle of 100m X 400m, there are two sides opposite each other that are 100m long and two sides next to them that are opposite each other that are 400m long. This means that you can work out the area if you only have the length of one side. To work out the area of a rectangle with a base length of 200m, I subtract 200m from 500m, giving me 300m and then times 200m by 300m.
I can then put this into an equation form: 1000 = X (500 -X)
X = Base Length
Below is a table of results, worked out using the formula above. I have gone down the table by taking 10m off the base every time.
There are three columns one showing the height (m) another showing the base length and the final column shows the area (m2).
Base Length
Height (m)
Area (m2)
500
0
0
490
0
4900
480
20
9600
470
30
4100
460
40
8400
450
50
22500
440
60
26400
430
70
30100
420
80
33600
410
90
36900
400
00
40000
390
10
42900
380
20
45600
370
30
48100
360
40
50400
350
50
52500
340
60
54400
330
70
56100
320
80
57600
310
90
58900
300
200
60000
290
210
60900
280
220
61600
270
230
62100
260
240
62400
250
250
62500
240
260
62400
230
270
62100
220
280
61600
210
290
60900
200
300
60000
90
310
58900
80
320
57600
70
330
56100
60
340
54400
50
350
52500
40
360
50400
30
370
48100
20
380
45600
10
390
42900
00
400
40000
90
410
36900
80
420
33600
70
430
30100
60
440
26400
50
450
22500
40
460
8400
30
470
4100
20
480
9600
0
490
4900
0
500
0
I can draw a graph of base length against area using this formula, (Fig 1.0).
According to the table and the graph, the rectangle with a base of 250m has the greatest area. As you can see in the table below I have gone into greater depth to find if the rectangle with a base of 250m is the highest possible area. This shape is also called a square, or a regular quadrilateral. Because I only measured to the nearest 10m, I cannot tell whether the graph is true, and does not go up just to the sides of 250m, so I will have to investigate further by focusing around 250m. I will work out the results using 249m, 249.5 and 249.75, as shown below.
Height (m)
Base (m)
Area (m2)
251
249
62499
250.5
249.5
62499.75
250.25
249.75
62499.9375
250
250
62500
249.75
250.25
62499.9375
249.5
250.5
62499.75
249
251
62499
All of these results fit into the graph line that I have, making my graph reliable, Fig 1.0.
