2. A Fencing Problem An Investigation

3. Fencing Problem A farmer has 315m of fencing. He also has a field with a large wall. He uses the wall and the fencing to close off a rectangular area as shown in the diagram. What is the largest rectangular area he can fence off using the wall and his fencing? x fencing wall Sheep

4. Let x be the length shown in the diagram. Obtain an expression for the area of grass available to the sheep. Enter the function for the area in Y 1 on your calculator. Set TBLSET to TblStart =10 and ∆Tbl = 10. Press 2nd Fn Table on the calculator and complete the table below for your results. x fencing wall Sheep

5. Table 1 xArea 102950 205500 307650 40 50 60 70 80 90

6. We now look closer between 70 and 90. Why? Set TBLSET to TblStart =70 and ∆Tbl = 2. Obtain a Table as before and enter your results. You should obtain a table like that shown in the next slide.

7. Table 2 xArea 7012250 7212312 7412358 7612388 7812402 8012400 8212382

8. Where is the maximum likely to occur now? Make up a third table using TBLSET = 76 and ∆Tbl = 1. You should now have the maximum area. What is the value for x which gives this maximum?

9. Repeat the previous calculations to find the maximum area for each of the examples which follow. Question 1. x fencing wall Sheep 417 metres of fencing

10. Question 2 Question 3 x fencing wall Pigs 183 metres of fencing x fencing wall Goats 229 metres of fencing

11. Question 4 This time some of the fencing is used so that separate compartments can be added for the goats, sheep and cows. Find the maximum possible total area. wall x fencing SheepPigsGoatsxx x 335m of fencing

12. Question 5 Four separate compartments in this example. Find the maximum total area. x fencing wall Sheep Pigs Goats x x x x Hens 415m of fencing