Answer
$C = 82^{\circ}26'$
$b = 283.2~m$
$c = 415.2~m$
Work Step by Step
We can find angle $C$:
$A+B+C = 180^{\circ}$
$C = 180^{\circ}- A -B$
$C = 180^{\circ}- 39^{\circ}54' -42^{\circ}32'$
$C = 82^{\circ}26'$
We can find the length of side $b$:
$\frac{b}{sin~B} = \frac{a}{sin~A}$
$b = \frac{a~sin~B}{sin~A}$
$b = \frac{(268.7~m)~sin~(42^{\circ}32')}{sin~(39^{\circ}54')}$
$b = \frac{(268.7~m)~sin~(42.53^{\circ})}{sin~(39.9^{\circ})}$
$b = 283.2~m$
We can find the length of side $c$:
$\frac{c}{sin~C} = \frac{a}{sin~A}$
$c = \frac{a~sin~C}{sin~A}$
$c = \frac{(268.7~m)~sin~(82^{\circ}26')}{sin~(39^{\circ}54')}$
$c = \frac{(268.7~m)~sin~(82.43^{\circ})}{sin~(39.9^{\circ})}$
$c = 415.2~m$
