An enclosed area is designed using one existing wall, and 72m of fence material. What is the maximum area of the enclosure?

The maximum area possible is 648 squared meters.

Step-by-step explanation:

Let the length of the existing wall be .

And let the width of the fence be .

The area of the enclosure will be given by:

Since the area is bounded by one existing wall, the perimeter (the 72 meters of fencing material) will only be:

We want to maximize the area.

From the perimeter, we can subtract 2w from both sides to obtain:

Substituting this for our area formula, we acquire:

This is now a quadratic. Recall that the maximum value of a quadratic always occurs at its vertex.

We can distribute:

Find the vertex of the quadratic. Using the vertex formula, we acquire that:

So, the maximum area is:

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