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Arcturus is about half as hot as the Sun but is about 100 times more luminous. What is its radius campared to the sun.

If Arcturus is half as hot, it radiates (1/2)^4 = 1/16 as much as the sun PER UNIT AREA. But since it radiates 100 times more totally, it must have 1600 times the area of the sun. That means it must have sqrt 1600 = 40 times the radius, since radius is proportional to the square root of area

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