Penning Pony

By:Allison K. Kootsikas


 

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).



The information from the problem states that one side of the fence is already there.This will affect how I solve the problem.The regular formula for perimeter is:(length x 2)+(width x 2).The formula for area is (length x width).Iím trying to find the largest possible area with the pen having a perimeter of 96 meters.

 

In this case, the formula for the perimeter will be

(width x 2)+length=96 meters


Click here for my GSP sketch.

 

I decided to investigate this using an Excel spreadsheet.

Length

Width 1

Width 2

Perimeter

(w*2)+l

Area

94

1

1

96

96

94

93

1.5

1.5

96

96

139.5

92

2

2

96

96

184

91

2.5

2.5

96

96

227.5

90

3

3

96

96

270

89

3.5

3.5

96

96

311.5

88

4

4

96

96

352

87

4.5

4.5

96

96

391.5

86

5

5

96

96

430

85

5.5

5.5

96

96

467.5

84

6

6

96

96

504

83

6.5

6.5

96

96

539.5

82

7

7

96

96

574

81

7.5

7.5

96

96

607.5

80

8

8

96

96

640

79

8.5

8.5

96

96

671.5

78

9

9

96

96

702

77

9.5

9.5

96

96

731.5

76

10

10

96

96

760

75

10.5

10.5

96

96

787.5

74

11

11

96

96

814

73

11.5

11.5

96

96

839.5

72

12

12

96

96

864

71

12.5

12.5

96

96

887.5

70

13

13

96

96

910

69

13.5

13.5

96

96

931.5

68

14

14

96

96

952

67

14.5

14.5

96

96

971.5

66

15

15

96

96

990

65

15.5

15.5

96

96

1007.5

64

16

16

96

96

1024

63

16.5

16.5

96

96

1039.5

62

17

17

96

96

1054

61

17.5

17.5

96

96

1067.5

60

18

18

96

96

1080

59

18.5

18.5

96

96

1091.5

58

19

19

96

96

1102

57

19.5

19.5

96

96

1111.5

56

20

20

96

96

1120

55

20.5

20.5

96

96

1127.5

54

21

21

96

96

1134

53

21.5

21.5

96

96

1139.5

52

22

22

96

96

1144

51

22.5

22.5

96

96

1147.5

50

23

23

96

96

1150

49

23.5

23.5

96

96

1151.5

48

24

24

96

96

1152

47

24.5

24.5

96

96

1151.5

46

25

25

96

96

1150

45

25.5

25.5

96

96

1147.5

44

26

26

96

96

1144

43

26.5

26.5

96

96

1139.5

42

27

27

96

96

1134

41

27.5

27.5

96

96

1127.5

40

28

28

96

96

1120

39

28.5

28.5

96

96

1111.5

38

29

29

96

96

1102

37

29.5

29.5

96

96

1091.5

36

30

30

96

96

1080

35

30.5

30.5

96

96

1067.5

34

31

31

96

96

1054

33

31.5

31.5

96

96

1039.5

32

32

32

96

96

1024

31

32.5

32.5

96

96

1007.5

30

33

33

96

96

990

29

33.5

33.5

96

96

971.5

28

34

34

96

96

952

27

34.5

34.5

96

96

931.5

26

35

35

96

96

910

25

35.5

35.5

96

96

887.5

24

36

36

96

96

864

23

36.5

36.5

96

96

839.5

22

37

37

96

96

814

21

37.5

37.5

96

96

787.5

20

38

38

96

96

760

19

38.5

38.5

96

96

731.5

18

39

39

96

96

702

17

39.5

39.5

96

96

671.5

16

40

40

96

96

640

15

40.5

40.5

96

96

607.5

14

41

41

96

96

574

13

41.5

41.5

96

96

539.5

12

42

42

96

96

504

11

42.5

42.5

96

96

467.5

10

43

43

96

96

430

9

43.5

43.5

96

96

391.5

8

44

44

96

96

352

7

44.5

44.5

96

96

311.5

6

45

45

96

96

270

5

45.5

45.5

96

96

227.5

4

46

46

96

96

184

3

46.5

46.5

96

96

139.5

2

47

47

96

96

94

1

47.5

47.5

96

96

47.5

 

After completing a spreadsheet, I see that the maximum area will be 1152 sq. meters.At this point, the length is 48 meters and the width is 24 meters.In the following graph, you can see the apex of the areas at 1152 sq. meters.

 

 

Click here to see my Excel spreadsheet.