If the Moon is 30 arc minutes (0.5 degrees) in diameter, how many Moon diameters would it take to span the distance from a point on the eastern horizon to a point directly opposite on the western horizon?
There are 360 degrees in a circle; therefore, one-half of a degree will fit into 360 degrees 720 times. If there are 720 "Moon diameters" in a circle, then half of those will cover half of a circle. 360 Moon diameters will cover an arc of 180 degrees.
If, by chance, this question is specifying the actual horizon as the opposite point, and not the 180 degree opposite of a point, then the answer is a bit more complicated, and depends on the perspective of the observer. The horizon changes with altitude; the higher the observer, the wider the angle of observation. However, for most human purposes the visible portion of the sky from one horizon to another will be around 180 degrees.
Arc minutes and arc seconds are units of measurement that we use to describe the apparent size of objects in the sky. They're a way of subdividing the degrees in a circle to make it easier to describe things that are very small, since even the largest solid objects in the sky (the moon and sun) don't even occupy 1 degree. An arc minute is 1/60th of a degree, and an arcsecond is 1/60th of 1 arcminute. Despite their names, they don't really have anything to do with time; they just share their mathematical proportional relationships with hours, minutes and seconds.