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# Graphing Calculator - Finding Maximum Area

So you have a couple hundred feet of fencing and you want to know the largest area you can enclose. No problem! This is a classic maximum area problem that is now very easy to solve with the help of a graphing calculator..

Just let each end of the rectangle equal X. There are two ends so that means you have two X’s.

Therefore, you have 200 – 2x left for the rest of the rectangle’s perimeter. But you have two sides so that means each side is 100 – x. This gives us the following diagram to use to think about the rest of the problem.

We all know that area = length x width.

Area = x(100 – x) = 100x – x^{2}

Now we could use some algebra to do the rest but these little tutorials are for non-math people so let’s make it easy. Plus, as one of my adult friends once said, I haven’t added x’s and y’s since I left high-school.

Instead, we are going to zip over to http://webgraphing.com/graphing_basic.jsp and use the nifty little graphing calculator to calculate the maximum area without anymore cumbersome algebra..

Plug the equation into the empty box at the top of the graphing calculator as follows and click on Graph It.

You’ll get the following parabola drawn before you can move your mouse again.

Voila….the top of the curve is the maximum area, 2500 square feet, and drop straight down for the value of x, which is 50 feet.

So the rectangle with the biggest area is 50’ x 50’.

Of course, you could probably guess the maximum area or figure it out by trial and error but this is a lot more fun!

**Up against a building**

What if our little fenced-in area is up against a building, you ask. Well, it’s not a lot more difficult. You still have the two X’s but now the remainder is just one side. The picture looks like this:

Again, the area is length x width:

Area = x(200 – 2x) = 200x –2 x^{2}

Put this one in the calculator and you will get the following graph.

This time our x value is 50 feet so a 50' x 100' pen gives us the biggest area of 5 000 square feet.

A lot more area for a lot less fence!