Sheep fencing problems

1
Sheep fencing problems

• Calculus

2
A farmer has a field in which there is a very
long straight wall. The farmer also has 340
metres of fencing and he wishes to use this with
the wall to fence of a rectangular area so that
he can keep his sheep in and prevent them from
wandering. The diagram below illustrates the
situation.
If x is the length of the short side of the
rectangle shown, show that the area A which can
be fenced is given by A x(340 2x). Use
calculus to find the dimensions of the field to
ensure that the area will be a maximum.
3
Repeat the above calculations for a farmer in
the same situation who has 246 metres of fencing.
4
A farmer has a field which contains two long
walls which meet at right angles as shown. He
also has 214 metres of fencing. Calculate the
largest rectangular area which he can fence off.
5
Repeat the above situation for a farmer who has
157 metres of fencing.
6
A farmer has 259 metres of fencing. He wishes to
fence off a rectangular area. Use calculus to
find the maximum possible area he can fence off
for his sheep.