Answer:
[tex]\boxed{\text{D) }C = 105 ^{\circ} 20', b = 45.91; c = 61.38}[/tex]
Step-by-step explanation:
A = 28° 30' = 28.5 °
B = 46° 10' =46.1667°
a = 30.37
1. Calculate the value of C
[tex]\begin{array}{rcl}A + B + C & = & 180\\28.5 + 46.1667 + C & = & 180\\74.6667 + C & = & 180\\C & = & 105.3333 ^{\circ} = \boxed{\mathbf{105 ^{\circ} 20'}}\\\end{array}[/tex]
Your triangle is an obtuse scalene triangle. It looks like the one below.
2. Calculate side b
Use the Law of Sines.
[tex]\begin{array}{rcl}\dfrac{b}{a } & = & \dfrac{\sin B }{\sin A }\\\\\dfrac{b}{30.37 } & = & \dfrac{\sin 46.1667 }{\sin 28.5 }\\\\\dfrac{ b}{30.37} & = & \dfrac{0.7214}{0.4772}\\\\\dfrac{ b}{30.37} & = & 1.512\\b & = & \boxed{\mathbf{45.91}}\\\\\end{array}[/tex]
3. Calculate side c
[tex]\begin{array}{rcl}\dfrac{c}{a } & = & \dfrac{\sin C }{\sin A }\\\\\dfrac{c}{30.37} & = & \dfrac{\sin 105.3333}{\sin 28.5}\\\\\dfrac{ c}{30.37} & = & \dfrac{0.9644}{0.4772}\\\\\dfrac{ c}{30.37} & = & 2.021\\c & = & \boxed{\mathbf{61.38}}\\\\\end{array}[/tex]
