a farmer has 500 metres of fencingg materials with whihc to enclose a rectangular paddock. He wishes to ensure that the paddock has the maxium area possible.

Question

construct a labelled diagram of the paddock and write the rule for its perimeter. Re-arrange the rule to express the length in terms of the width. State the rule for the area of the paddock in terms of the width only.

Sketch the graph of the area versus width. you may do using technology

Using your graph, find the maximum area of the paddock, and state the dimensions which would generate this area. What shape is this paddock?

Answer 1

w represents width

250 - w represents length

 

A = w(250 - w)

A = 250w - w²

A = -w² + 250w

 

Maximum is at -250/(2(-1)), or 125

 

The rectangle with th greatest area for a given perimeter is the square.