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Interested in mathematics!
(1) Solution: Let the length of the wooden fence perpendicular to the wall be x meters.

Get: (23-3X)*X=40 and: 23-3X.

The solution is X=5.

(The chicken farm is rectangular, 8 meters long and 5 meters wide)

(2) no.

Suppose you can design a chicken farm with an area of 50 square meters. Assume that the length of the wooden fence perpendicular to the wall is y meters.

Get: (23-3Y)*Y=50

3Y^-23Y+50=0

It is easy to get δ = 23-4 * 3 * 50 =-79.

So this equation doesn't hold. You can't design a chicken farm with an area of 50 square meters.