Penning for Pony

By Page Bird

 

To make a pen for his new pony, Ted will use an existing fence as one side of the pen. If he has ninety-six meters of fencing, what are the dimensions of the largest rectangular pen he can make?

(Source: Mathematics Teaching in the Middle School, Nov-Dec1994).

 


First, I made a picture using GSP that represents the pen for the pony:

 

 

 



The line in the picture represents the side of the pen that is already given.  Since the pen is rectangular, opposite sides are equal.  Therefore, sides AB and DC have the same length.  Looking at the picture below, I will label side AB and side DC as x in order to show that they have the same length.  I will represent the length of side AD with y.

 


We know that the perimeter of the three sides can be found by adding the side lengths.  Therefore the distance of the three sides is

 

x + x + y =  2x + y = 96 meters.

 

So the distance of y is the same as 96 – 2x, i.e., y = 96 – 2x.

 

Since we are looking for the dimensions of the largest rectangular area, I am going to use a spreadsheet to find these dimensions. 

But first, the area can be found by multiplying the dimensions of the rectangle.  That is, area equals the length times the width.  So we can say,

 

Area = x * y.  And since we let y = 96- 2x, then Area = x * (96-2x)

 

I will plug this equation into the spreadsheet.  When I do, I get the following results:

 

length

width

Area

x

96 - 2x

x* (96-2x)

1

94

94

2

92

184

3

90

270

4

88

352

5

86

430

6

84

504

7

82

574

8

80

640

9

78

702

10

76

760

11

74

814

12

72

864

13

70

910

14

68

952

15

66

990

16

64

1024

17

62

1054

18

60

1080

19

58

1102

20

56

1120

21

54

1134

22

52

1144

23

50

1150

24

48

1152

25

46

1150

26

44

1144

27

42

1134

28

40

1120

29

38

1102

30

36

1080

31

34

1054

 

This spreadsheet shows that the greatest area is 1,152 m2.  When the area is 1, 152 m2 , the length is 24 meters and the width is 48 meters. 

 

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