Meaning and origin of "at bay"

Solution 1:

Etymonline links this to howling dogs

bay (n.3)

"howl of a dog," early 14c., earlier "howling chorus raised (by hounds) when in contact with the hunted animal," c.1300, from O.Fr. bayer, from PIE base *bai- echoic of howling (cf. Gk. bauzein, L. baubari "to bark," English bow-wow; cf. also bawl).

From the hunting usage comes the transferred sense of "final encounter," and thence, on the notion of putting up an effective defense, at bay. As a verb, "to bark or howl (at)," from late 14c.

An interesting explanation is found on the Word Detective archives

It's that last, "howling dog," sense of "bay" that gave us "at bay." When "bay" first appeared in this sense in English around 1300, it meant the chorus of howling barks of a pack of hunting hounds in hot pursuit of their prey. (This "bay" as "howling bark" also is used in the phrase "baying at the moon," which some dogs lacking access to television are known to do.)

In the final stage of such a grim chase, the hunted animal (fox, raccoon, etc.) will often find itself cornered and turn to face the pursuing pack. If the animal is sufficiently determined (and who wouldn't be?), it may be able to at least temporarily repel the dogs, rendering them able only to stand and howl, thus keeping them "at bay."

As a metaphor for holding off difficulties (and making a perhaps ultimately futile last stand), "to hold at bay" was in use applied to humans and their problems by the 16th century.