1. A Fencing Problem
An Investigation

2. A farmer has 315m of fencing.
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

3. 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 Y1 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

4. Table 1 x
Area 10
2950 20
5500 30
7650 40 50 60 70 80 90

5. 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.

6. Table 2 x
Area 70
12250 72
12312 74
12358 76
12388 78
12402 80
12400 82
12382

7. 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?

8. 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

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

10. 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
Sheep
Pigs
Goats
335m of fencing

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