A Fencing Problem

An Investigation

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

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

Table 1x Area

10 2950

20 5500

30 7650

40

50

60

70

80

90

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.

Table 2x Area

70 12250

72 12312

74 12358

76 12388

78 12402

80 12400

82 12382

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?

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

Question 2

Question 3

xfencing

wall

Pigs

183 metres of fencing

xfencing

wall

Goats

229 metres of fencing

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 PigsGoats

xx x

335m of fencing

Question 5

Four separate compartments in this example.

Find the maximum total area.

x

fencing

wall

Sheep PigsGoats x

x xxHens

415m of fencing