Sheep fencing problems

Calculus

wall

sheep

fencing

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.

Repeat the above calculations for a farmer in the same situation who has 246 metres of fencing.

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.

fencing

wall

wall

sheep

Repeat the above situation for a farmer who has 157 metres of fencing.

fencing

sheep

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.