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A square plot of land feet on each side is shown in the diagram at right. A pipe is to be laid in a straight line from point to a point on , and from there to point . (Point cannot be point or point .) The cost of laying the pipe is per foot if it goes through the lot from point A to point P (since it must be laid underground), and per foot if it is laid along one of the sides of the lot. What is the most economical way to lay the pipe?

Let .
Now write expressions for and .
The function for the cost is .
Once you have written the cost function in terms of , differentiate and determine the value of that yields a minimum.

Note: This is not the answer to the question being asked.