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 m^{2}.
When the area is 1, 152 m^{2} , the length is 24 meters and the
width is 48 meters.

Click here to return to my home page.