Question

The lifetime of a star is rougly inversly propotional to the cube of its mass. Our sun, which has a mass of 1 solar mass, will last for approxamatly 10 billion years.

How long will a star that is half as massive as the sun last?

Answer 1

Let T be the lifetime and M the mass. You're told that the lifetime, T, is roughly inversely proportional to the cube of the mass, M; that is,\[T \approx \frac{k}{M^3}\]where k is some constant of proportionality.

We can compare, then, the lifetime of two stars as, \[\frac{T_2}{T_1}\approx \frac{\frac{k}{M_1^3}}{\frac{1}{M^2_3}}=\frac{M_1^3}{M^3_2}\]From the information, you have that the mass of the other star will be \[M_{Star}=\frac{1}{2}M_{sun}\]so,\[\frac{T_{Star}}{T_{sun}}\approx \frac{M^3_{sun}}{(\frac{M_{sun}}{2})^3}=2^3=8 \]So the lifetime of the star is approximately,\[T_{Star}\approx 8T_{sun}=80 \times 10^9yr\]that is, 80 billion years.

Larger stars consume their fuel faster than smaller stars, so this result makes sense.