To find the length of side $a$ using the Law of Sines, we use the following formula:

$\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}$

Given: $c = 8 \text{ mi}$ $A = 107.73^\circ$ $B = 39.78^\circ$ $C = 32.49^\circ$

We need to find $a$. Using the Law of Sines, we can write:

$\frac{a}{\sin(A)} = \frac{c}{\sin(C)}$

Plugging in the known values:

$\frac{a}{\sin(107.73^\circ)} = \frac{8}{\sin(32.49^\circ)}$

We need to find the sine values:

$\sin(107.73^\circ) \approx 0.9563$ $\sin(32.49^\circ) \approx 0.5360$

Now, plug these into the equation:

$\frac{a}{0.9563} = \frac{8}{0.5360}$

Solving for $a$:

$a = \frac{8 \cdot 0.9563}{0.5360} \approx \frac{7.6504}{0.5360} \approx 14.28 \text{ mi}$

Rounding to the nearest mile:

$a \approx 14 \text{ mi}$

Therefore, the length of side $a$ is approximately: A. $a = 14$ mi

Would you like any more details or have any questions about the process?

Here are 5 questions you might consider asking:

1. How do you find the length of side $b$ in this triangle?
2. Can you explain how to solve a triangle using the Law of Cosines?
3. What are the steps for solving a triangle given all three sides?
4. How can you determine if a triangle is valid based on its angles and sides?
5. What are the common mistakes to avoid when using the Law of Sines?

Tip: When solving trigonometric problems, make sure your calculator is set to the correct mode (degrees or radians) based on the given angles.