A track is for mice, designed in the shape of a rectangle, sandwiched between two semicircles. If the track must have a fixed length of exactly 4 feet, what is the largest area you can enclose with this track?

Let one side be x.  Since there are two sides of this length, that makes up 2x.
That leaves 4-2x for the sum of the other two sides.
This makes one of the other sides 2-x.
With length x and width 2-x, the volume is x(2-x) = 2x - x^2.

If we take f(x) = 2x - x^2, then f'(x) = 2 - 2x.
Setting this to 0 gives 2 - 2x = 0.
Adding 2x to both sides gives 2 = 2x.
Dividing this by 2 gives x = 1.

Since this is the length of one side, and there are two sides, this has a total length of 2.
This leaves 4-2 = 2 for the total of the other two sides.  Since there are two sides for this,
dividing by 2 gives 2/2 = 1.  This means the other sides has length 1.

Since the first side has length 1 and width 1, it is a square of length 1 on each side.

This gives an area of 1x1 = 1.


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