What is the largest rectangular area of land you could fence off? 99 cm2
(With a 40 metre fence)
You can find the perimeter of a rectangle by adding the numbers in the formula for area (l x w) and then multiply that by two.
For example
Area = 2 x 3
Perimeter = 2(2+3)
Or. 2+2+3+3
= 10
Therefore to get a perimeter of 40 every time the length and width (1 of each) has to add up to 20. Hence when doubled will equal 40.
19 x 1 = 19cm2
18 x 2 = 36cm2
17 x 3 =51cm2
16 x 4 =64cm2
15 x 5 =75cm2
14 x 6 =84cm2
13 x 7 =91cm2
12 x 8 =96cm2
11 x 9 =99cm2
Obviously 10 x 10 being 100 but making a square not a rectangle (equal length and width).
What is the largest rectangular area of land you could fence off? 200 cm2
The wall is covering one of the length sides therefore we need two of the same number add another number, bigger than those being multiplied (2 widths + one bigger length) to make 40
Eg. 1+1+38=40
1+1+38=40
1 x 38 = 38cm2
2+2+36=40
2 x 36= 72
3+3+34=40
3 x 34 = 102
4+4+32=40
4 x 32 = 128
5+5+30=40
5 x 30 = 150
6+6+28=40
6 x 28= 168
7+7+26=40
7 x 26=182
8+8+24=40
8 x 24=192
9+9+22=40
9 x 22=198
10+10+20=40
10 x 20= 200cm2
However if we continue with the same process, we see that areas reduce again...
11+11+18=40
11 x 18= 98
12+12+16=40
12x16=192
13+13+14=40
13 x 14=182
14+14+12= 40. However the smaller number cannot work as the second number has to be larger as it is the length
What is the largest area you can fence off now?
With the large wall , it is easier to section off the 8x10 rectangle. Seeing as two of the sides are covered by the wall and the other 8m is in the middle of the overall rectangle only 10 meters of fence is taken up on that sectioned off rectangle.
8 x 10= 80 cm2
10m fence
So with 10 metres gone for the portioned off second rectangle, that leaves us with 30 metres of fence left. We do the same process as we did for the first question but with 30 as the perimeter allowance.
1 x 14=14cm2
2 x 13=26
3 x 12=36
4 x 11=44
5 x 10=50
6 x 9= 54
7 x 8=56
The last one is the biggest area but you cannot go any further than this by multiplication as it reverses back on itself eg. 8x7, 9 x 6 .....
80cm2 + 56cm2= 136cm2 with 40 metres of fencing used
Extension
I do not believe flexible, curvy fencing would create larger areas fenced off. Even if an area was created in a circle, that would not be as big, in area as a square.