Maximum Area Field with a Corner Wall
Maximum Area Field with a Corner Wall
L-x
A farmer has an by foot corner wall and feet of fence. He wants to use the fence to construct an by foot rectangular field using the corner wall for one corner, and part or all of its sides. What should and be to maximize the area of the field?
a
b
L
x
y
x
y
A=xy
For the demonstration, , , , and . There are four cases relating , , , and , depending on whether , and , and , or . Vary to see the possible field boundaries.
0≤a≤100
0≤b≤100
1≤L≤4Max(1,a,b)
0≤x≤MinL,
(L+a)
2
x
y
a
b
L
0≤x,y≤Min(a,b)
0≤x≤a
b≤y
a≤x
0≤y≤b
x,y≥Max(a,b)
x
The largest field is shown in green, and as a "double square" when it is a 2 by 1 or 1 by 2 rectangle. Study its shape as (or or ) varies, in the cases , ≤≤2 and . It can be a square, a 1x2 or 2x1 rectangle, or an "intermediate" rectangle. The possible shapes and a formula for the dimensions of the largest field are derived in the Details.
L
a
b
0≤≤
b
a
1
2
1
2
b
a
2≤
b
a