Aristarchus measured the Sun-Earth-Moon angle at quarter moon to be 87° and calculated that the Sun is 20 times as far away as the Moon. Actually, the Sun is almost 400 times as far away as the Moon. Thus the actual Sun-Earth-Moon angle at quarter moon must be:
A)less than 45°.
B)45°.
C)86.5°.
D)89.5°.

Question

Quizplus · Accepted Answer

Since Aristarchus believed the Sun was only 20 times as far away as the Moon, he underestimated the actual distance by a factor of 400/20 = 20. Therefore, the angle must be larger than what he calculated in order to account for this discrepancy. If we assume the Moon is at a right angle to the Earth-Sun line during quarter moon (which is a reasonable assumption), then the Sun-Earth-Moon angle at quarter moon would be 90°. Thus, we can estimate that the actual angle is slightly less than 90°, which would make choice D, 89.5°, the best option.

Aristarchus Measured the Sun-Earth-Moon Angle at Quarter Moon to Be