Answer:
x = 28.2
Step-by-step explanation:
To find the value of x, first find the measure of side AC using the Law of Cosines, then use the Law of Sines to find the measure of angle C.
[tex]\boxed{\begin{array}{l}\underline{\textsf{Law of Cosines}}\\\\b^2=a^2+c^2-2ac \cos B\\\\\textsf{where:}\\\phantom{ww}\bullet\;\textsf{$a, b$ and $c$ are the sides.}\\\phantom{ww}\bullet\;\textsf{$B$ is the angle opposite side $b$.}\end{array}}[/tex]
In this case:
[tex]a = BC = 12.3\\\\b = AC\\\\c = AB = 9.7\\\\B = 115^{\circ}[/tex]
Substitute the values into the formula and solve for the exact measure of AC:
[tex]AC^2=12.3^2+9.7^2-2(12.3)(9.7)\cos 115^{\circ}\\\\\\AC^2=151.29+94.09-238.62\cos 115^{\circ}\\\\\\AC^2=245.38-238.62\cos 115^{\circ}\\\\\\AC=\sqrt{245.38-238.62\cos 115^{\circ}}[/tex]
[tex]\boxed{\begin{array}{l}\underline{\textsf{Law of Sines}} \\\\\dfrac{\sin A}{a}=\dfrac{\sin B}{b}=\dfrac{\sin C}{c}\\\\\textsf{where:}\\\phantom{ww}\bullet \;\textsf{$A, B$ and $C$ are the angles.}\\\phantom{ww}\bullet\;\textsf{$a, b$ and $c$ are the sides opposite the angles.}\end{array}}[/tex]
in this case:
[tex]B = 115^{\circ}\\\\b = AC=\sqrt{245.38-238.62\cos 115^{\circ}}\\\\C = x^{\circ}\\\\c = AB=9.7[/tex]
Substitute the values into the formula and solve for x:
[tex]\dfrac{\sin 115^{\circ}}{\sqrt{245.38-238.62\cos 115^{\circ}}}=\dfrac{\sin x^{\circ}}{9.7}\\\\\\\\\sin x^{\circ}=\dfrac{9.7\sin 115^{\circ}}{\sqrt{245.38-238.62\cos 115^{\circ}}}\\\\\\\\x^{\circ}=\sin^{-1}\left(\dfrac{9.7\sin 115^{\circ}}{\sqrt{245.38-238.62\cos 115^{\circ}}}\right)\\\\\\\\x^{\circ}=28.1943172...^{\circ}\\\\\\x^{\circ}=28.2^{\circ}\\\\\\x=28.2\; \sf (3\;s.f.)[/tex]
Therefore, the value of x correct to 3 significant figures is:
[tex]\Large\boxed{\boxed{x=28.2}}[/tex]
