Penning the Pony
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?
First, make a sketch of your knowns in Geometry Sketchpad:
I know that the sum of the3 sides must be equal to 96 meters. I am looking to maximize the product of side 1 and 2.
For efficiency, let side 1= x, side 2 =y and side 3 = z.
In mathese my sentence above is x + y + z = 96.
Since side 3 is equal to side 1, (because this is specified to be a rectangle) I can substitute x for z.
Therefore 2x + y = 96
Next I will go to a spreadsheet to look at table of areas.
Set up the first column as side 1 and fill it down with a series starting with 1.
Set up the second column as side 2. Solving the equation above for y, insert the formula 96-2*A1. Lastly set up column 3 as the area. So insert the formula A2*B2.
As you peruse the table, notice that the largest area is 1152 feet squared so the dimensions of side 1 is 24 feet and side 2 is 48 feet.
You can also see this as a graph. Since y= 96-2x as determined by the perimeter. You can plug this into an area formula so y=x(96-2x)