# 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.