Answer
$$C = {97^ \circ }34',\,\,\,\,\,\,b = 283.18{\text{m}},\,\,\,\,\,\,\,c = 415.24{\text{m}}$$
Work Step by Step
$$\eqalign{
& A = {39^ \circ }54',\,\,\,a = 268.7{\text{m,}}\,\,\,B = {42^ \circ }32' \cr
& \cr
& {\text{Find }}C \cr
& C = {180^ \circ } - A - B \cr
& C = {180^ \circ } - {39^ \circ }54' - {42^ \circ }32' \cr
& C = {97^ \circ }34' \cr
& \cr
& {\text{Use the law of sines to find side }}b \cr
& \frac{a}{{\sin A}} = \frac{b}{{\sin B}} \cr
& b = \frac{{a\sin B}}{{\sin A}} \cr
& b = \frac{{268.7\sin \left( {{{42}^ \circ }32'} \right)}}{{\sin \left( {{{39}^ \circ }54'} \right)}} \cr
& {\text{Use a calculator}} \cr
& b = 283.18{\text{m}} \cr
& \cr
& {\text{Use the law of sines to find side }}c \cr
& \frac{c}{{\sin C}} = \frac{a}{{\sin A}} \cr
& c = \frac{{a\sin C}}{{\sin A}} \cr
& c = \frac{{268.7\sin \left( {{{97}^ \circ }34'} \right)}}{{\sin \left( {{{39}^ \circ }54'} \right)}} \cr
& {\text{Use a calculator}} \cr
& c = 415.24{\text{m}} \cr
& \cr
& {\text{Answer}} \cr
& C = {97^ \circ }34',\,\,\,\,\,\,b = 283.18{\text{m}},\,\,\,\,\,\,\,c = 415.24{\text{m}} \cr} $$
