A star is half as hot as the sun but 100 times more luminousWhat is its radius to the sun?

Answer 1

The question, as General Haig was wont to say, "lacks proper texture." A star of solar class but ten times solar radius would be 100 times more luminous, but "half as hot" is a problematic qualifier because stars radiate ALL wavelengths of energy, not just heat or light. A star could be below average in infrared (heat) emission but about average in visible light, throwing off any simple calculations. To more accurately compare stellar radiation, we use a concept called "bolometric" measurement, which attempts to account for all forms of EM emission.

Another answer:

Surface area and surface temperature are the two factors involved in a star's energy output per second, its luminosity.

The energy radiated by a star is proportional to the fourth power of the star's absolute temperature. That's its temperature in Kelvin.

So, the luminosity of a star is proportional to its surface area (which is proportional to the square of its radius) and the fourth power of its temperature.

So the (slightly simplified) answer to the question can easily be worked out from there, thus:

The star is half as hot as the Sun. That would make its luminosity one sixteenth that of the Sun's.

But it is 100 times as luminous as Sun.

So, its surface area must be 1600 times the Sun's.

So, the star's radius must be 40 times the Sun's radius.