Answer
(a) $1.1\times 10^{31}J/K$
(b) $3.28\times 10^{31}J$
Work Step by Step
(a) We can find the required entropy increase as follows:
$Q_{sun}=3.80\times 10^{26}\times 3600\times 24$
$Q_{sun}=3283.2\times 10^{28}J$
Now $\Delta S=\frac{Q_{sun}}{T_{sun}}+\frac{Q_{sun}}{T_{universe}}$
We plug in the known values to obtain:
$\Delta S=\frac{3283.2\times 10^{28}}{5500+273}+\frac{3283.2\times 10^{28}}{3+273}$
$\Delta S=1.1\times 10^{31}J/K$
(b) We can find the required work done as
$\eta_{carnot}=1-\frac{T_{cold}}{T_{hot}}$
We plug in the known values to obtain:
$\eta_{carnot}=1-\frac{3+273}{5500+273}=0.95$
Now $W=\eta_{carnot}\times Q_{sun}$
We plug in the known values to obtain:
$W=0.95\times 3283.2\times 10^{28}$
$W=3.28\times 10^{31}J$
