Answer
$$a = 43.7{\text{km}},{\text{ }}B = {53.14^ \circ },\,\,C = {59.56^ \circ }$$
Work Step by Step
$$\eqalign{
& A = {67.3^ \circ },\,\,\,b = 37.9{\text{km,}}\,\,\,c = 40.8{\text{km}} \cr
& {\text{Use the Law of cosines to find }}a \cr
& {a^2} = {b^2} + {c^2} - 2bc\cos A \cr
& {\text{Substitute}} \cr
& {a^2} = {\left( {37.9} \right)^2} + {\left( {40.8} \right)^2} - 2\left( {37.9} \right)\left( {40.8} \right)\cos \left( {{{67.3}^ \circ }} \right) \cr
& {\text{Use a calculator}} \cr
& {a^2} \approx 1907581537 \cr
& {\text{Take square roots and choose the positive root}} \cr
& a \approx 43.7{\text{km}} \cr
& \cr
& {\text{Calculate the angle }}B{\text{ using the law of sines}} \cr
& \frac{b}{{\sin B}} = \frac{a}{{\sin A}} \cr
& \sin B = \frac{{b\sin A}}{a} \cr
& \sin B = \frac{{37.9\sin \left( {{{67.3}^ \circ }} \right)}}{{43.7}} \cr
& {\text{Use a calculator}} \cr
& \sin B \approx 0.8 \cr
& {\text{Use the inverse sine function}} \cr
& B \approx {\sin ^{ - 1}}\left( {0.8} \right) \cr
& B \approx {53.14^ \circ } \cr
& \cr
& {\text{Calculate }}C \cr
& C = {180^ \circ } - A - B \cr
& C = {180^ \circ } - {67.3^ \circ } - {53.14^ \circ } \cr
& C = {59.56^ \circ } \cr
& \cr
& {\text{Answer}} \cr
& a = 43.7{\text{km}},{\text{ }}B = {53.14^ \circ },\,\,C = {59.56^ \circ } \cr} $$
