A farmer has 110 metres of fencing. He wishes to build a rectangular paddock for grazing his sheep. Before he does it he draws up a couple of plans for two possible paddocks.

He notices that the first planned paddock has a smaller area than the second paddock. He wonders if he could plan a paddock with an even greater area (for more grass!).

Questions

1. Can you design a rectangular paddock from 110 metres of fencing with a biggerarea?

2. Look up the word optimisation in a dictionary or encyclopedia and show its meaning in a sentence or two.

3. Write down a formula for the perimeter (P) of a rectangular paddock with width(W) and length (L). What is the perimeter for each of the paddocks above.

4. Draw up a table like this one: 

5. Fill in the missing values. Do you notice anything about the numbers in the area(A) column?

6. Extend the table to include other suitable numbers that will help you to find the maximum possible area of a paddock using 110 metres of fencing. Is the best length (L) a whole number?

7. What do you think is the MAXIMUM area possible using only 11 metres of fencing?

8. By plotting points (L, A) from your table in questions 4, 5 and 6, draw a graph ofarea (A) versus length (L).Plot L on the horizontal axis. Show on the graph the maximum area A and the best value of L

Extension

A. If there was no restriction on the shape of the paddock, e.g. It could be a triangle, circle or trapezium, can you find a shape that encloses more area than the optimum area you found before.

B. Investigate the reverse situation. For example, if you need a paddock with an area of 200 square metres, what is the minimum amount of fencing you would need if:

a. The garden is a rectangle.
b. There is no restriction on the shape of the garden.