Solve the triangle.
\[
a=8.817 \text { in } c=6.072 \text { in } B=70.34^{\circ}
\]
What is the length of side $b$ ?
in
(Round to the nearest thousandth as needed.)
What is the measure of angle $A$ ?
(Round to the nearest hundredth as needed.)
What is the measure of angle $\mathrm{C}$ ?
(Round to the nearest hundredth as needed.)

Step 1 ：We are given two sides and an included angle of a triangle. We can use the Law of Cosines to find the length of side $b$. The Law of Cosines states that for any triangle with sides of lengths $a$, $b$, and $c$ and angles $A$, $B$, and $C$ opposite those sides, the following relationship holds: \[c^2 = a^2 + b^2 - 2ab\cos(C)\]

Step 2 ：We can rearrange this formula to solve for $b$: \[b = \sqrt{a^2 + c^2 - 2ac\cos(B)}\]

Step 3 ：After finding $b$, we can use the Law of Sines to find angles $A$ and $C$. The Law of Sines states that the ratio of the length of a side of a triangle to the sine of the angle opposite that side is the same for all three sides of the triangle. This gives us two equations: \[\frac{a}{\sin(A)} = \frac{b}{\sin(B)}\] and \[\frac{c}{\sin(C)} = \frac{b}{\sin(B)}\]

Step 4 ：We can rearrange these to solve for $A$ and $C$: \[A = \arcsin\left(\frac{a\sin(B)}{b}\right)\] and \[C = \arcsin\left(\frac{c\sin(B)}{b}\right)\]

Step 5 ：Substituting the given values $a = 8.817$, $c = 6.072$, and $B = 70.34$ into the formulas, we get $b = 8.864820307838514$, $A = 69.4923993310674$, and $C = 40.16760066893258$

Step 6 ：Rounding to the nearest thousandth for $b$ and to the nearest hundredth for $A$ and $C$, we get the final answer: The length of side $b$ is approximately \(\boxed{8.865}\) in. The measure of angle $A$ is approximately \(\boxed{69.49}\) degrees. The measure of angle $C$ is approximately \(\boxed{40.17}\) degrees.