Penning the Pony

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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?

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.

So, x+y+x=96

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)