$[3]($a$)$ If you burn a kilogram of gasoline, you get about Eg $=3\mathrm{.}6107$ Joules of energy. The

mass of the Sun is about M$=21030$ kg$.$ If you assume $($incorrectly$!)$ that the Sun is made

of gasoline, how long can it burn? This assumption is roughly equivalent to assuming that the

hydrogen in the Sun burns with the same energy per unit mass as listed above for gasoline,

i$.$e$.,$ that the Sun lives off of chemical energy. The power output of the Sun, also called its

luminosity, is L $=41026$ Watts $=41026$ Joule$/$sec$.$ Express your answer for the Sun$$s

lifetime in years. $($b$)$ Now assume that the Sun is powered by gravitational contraction. The

gravitational potential energy is given by the expression from problem $[2].$ Assume that the

solar radius R$=7108$ m$,$ and that it can contract by a factor of $10.$ For the given solar

luminosity, what is the lifetime of the Sun $($in years$)?$