What is the maximum area for this play zone if it is to fit into a right-triangular plot with sides measuring?

Question

Please show equation used and steps.
You need to fence in a rectangular play zone for children. What is the maximum area for this play zone if it is to fit into a right-triangular plot with sides measuring 4 meters by 12 meters?
If a graph was made with area function as a function of x( where x represents the length of the play zone. How would this be done?
Please show equation used and steps.

Answer 1

I answered a very similar question recently.

The basic idea is that if you draw a picture, there are a whole lot of similar triangles.

If x is the length and y is the height, assuming the side of length x is parallel to the side of length 12,

12/4 = (12-x)/y and also 12/4 = x/(3-y).

I think.

If not, some equations similar to those are correct. (Draw the picture -- which I didn't bother to do since I'm not the one with a homework assignment here -- and it should be obvious.)

Solving for x in terms of y shouldn't be hard.

And then the area is a function of x, and you can maximize using the usual techniques.