To find the distance from point $K$ to point $M$, we can apply the Cosine Rule in the triangle $KLM$. The Cosine Rule states:

$c^2 = a^2 + b^2 - 2ab \cdot \cos(C)$

where:

• $c$ is the side opposite angle $C$,
• $a$ and $b$ are the other two sides,
• $C$ is the included angle between sides $a$ and $b$.

In this case:

• $a = 180$ miles (from $K$ to $L$),
• $b = 203$ miles (from $L$ to $M$),
• $C = 107^\circ$ (angle at point $L$).

Substituting these values into the formula:

$KM^2 = 180^2 + 203^2 - 2 \times 180 \times 203 \times \cos(107^\circ)$

First, calculate the values step-by-step:

1. $180^2 = 32400$
2. $203^2 = 41209$
3. Calculate $\cos(107^\circ)$. Using a calculator, $\cos(107^\circ) \approx -0.2924$.
4. Calculate $2 \times 180 \times 203 \times (-0.2924) \approx -21249.84$.

Now substitute these values back:

$KM^2 = 32400 + 41209 + 21249.84$ $KM^2 \approx 94858.84$

Now, taking the square root to find $KM$:

$KM \approx \sqrt{94858.84} \approx 308 \text{ miles}$

Thus, the distance from $K$ to $M$ is approximately 308 miles.

Would you like further clarification on any step, or have any questions?

Here are some related questions:

1. How is the Cosine Rule derived and when is it typically applied?
2. What are some real-life applications of the Cosine Rule?
3. How would the result change if the angle was different?
4. Can we solve this problem using any other trigonometric approach?
5. How does rounding at each step affect the final answer?

Tip: When using trigonometric functions in calculations, keeping the intermediate values as precise as possible can minimize rounding errors in the final answer.