A certain pond absorbs I $=955$ W$/$m$2$ for a time period of an hour from the sun. Assume that all of this energy is used to raise the temperature of the pond. The pond water fills a hemisphere of radius r $=3\mathrm{.}5$ m and has a density $=1000$ kg$/$m$3.$ The specific heat of the pond water is c $=4186$ J$/($kg$$K$).$ Part $($a$)$

What is the change in temperature $$T in kelvins of the pond due to this energy transfer? $$ Part $($b$)$

What is the change in the internal energy of the pond $$U in joules, assuming that it does no work and loses no energy as heat?

$$ Part $($c$)$

If the initial temperature of the pond was T $=20\mathrm{.}0$C$,$ what would be the change in the entropy $$S of the pond, in joules per kelvin?

$$ Part $($d$)$

If the pond were at a temperature of T $=0\mathrm{.}0$C and covered by a d $=0\mathrm{.}10$ m thick layer of ice at this same temperature, how long $$t in hours would it take to melt the ice? The latent heat of fusion for ice is Lf $=334$ kJ$/$kg and its density is ice $=917$ kg$/$m$3.$ Assume that the layer of ice has the form of a cylinder of thickness d and cross$-$sectional radius r$.$

$$ Part $($e$)$

If the thickness of the ice were doubled, would it take twice as long to melt the ice?

$$ Part $($f$)$

If the radius of the pond were doubled, would it take longer to melt the ice?