Question: Solve for $c$. Given angles $B = 27^\circ$, $A = 34^\circ$, side $b = 11$, and using the Law of Sines: $\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}$. Find $C$ and then $c$. Round your answer to the nearest hundredth.
1. **State the problem:** We are given a triangle with angles $A = 34^\circ$, $B = 27^\circ$, and side $b = 11$. We need to find side $c$ opposite angle $C$, and angle $C$ itself.
2. **Find angle $C$:** The sum of angles in a triangle is $180^\circ$.
$$ C = 180^\circ - A - B = 180^\circ - 34^\circ - 27^\circ = 119^\circ $$
3. **Use the Law of Sines:**
$$ \frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c} $$
We want to find $c$, so use:
$$ \frac{\sin B}{b} = \frac{\sin C}{c} $$
Rearranged to solve for $c$:
$$ c = \frac{b \sin C}{\sin B} $$
4. **Calculate $c$:**
$$ c = \frac{11 \times \sin 119^\circ}{\sin 27^\circ} $$
Calculate the sines:
$$ \sin 119^\circ \approx 0.8746 $$
$$ \sin 27^\circ \approx 0.45399 $$
So:
$$ c = \frac{11 \times 0.8746}{0.45399} $$
5. **Simplify the fraction:**
$$ c = \frac{\cancel{11} \times 0.8746}{0.45399} = \frac{9.6206}{0.45399} $$
6. **Divide:**
$$ c \approx 21.19 $$
**Final answer:**
Side $c$ is approximately $21.19$ units.