In ΔLMN, m = 450 inches, mm∠M=74° and mm∠N=41°. Find the length of l, to the nearest 10th of an inch.

We are given a triangle \( \Delta LMN \) with the following information:
- Side \( m = 450 \) inches (the length opposite \( \angle M \)),
- \( \angle M = 74^\circ \),
- \( \angle N = 41^\circ \).

We are tasked with finding the length of side \( l \), which is opposite \( \angle L \).

### Step 1: Calculate \( \angle L \)
Using the fact that the sum of angles in a triangle is always \( 180^\circ \), we can calculate \( \angle L \):
\[
\angle L = 180^\circ - \angle M - \angle N = 180^\circ - 74^\circ - 41^\circ = 65^\circ.
\]

### Step 2: Apply the Law of Sines
We use the Law of Sines to relate the sides of the triangle to its angles:
\[
\frac{l}{\sin M} = \frac{m}{\sin L}.
\]
Substituting the known values:
\[
\frac{l}{\sin 74^\circ} = \frac{450}{\sin 65^\circ}.
\]

### Step 3: Solve for \( l \)
We can now solve for \( l \):
\[
l = \frac{450 \cdot \sin 74^\circ}{\sin 65^\circ}.
\]

Using a calculator to find the sine values:
\[
\sin 74^\circ \approx 0.9613, \quad \sin 65^\circ \approx 0.9063,
\]
we get:
\[
l = \frac{450 \cdot 0.9613}{0.9063} \approx \frac{432.585}{0.9063} \approx 477.4 \text{ inches}.
\]

Thus, the length of \( l \) is approximately **477.4 inches**.

Would you like further clarification on any part of the solution?

### Related questions:
1. How would the result change if \( m \) was different, for instance, 500 inches?
2. What if \( \angle N \) was larger, say \( 50^\circ \)? How would that affect the length of \( l \)?
3. Could we use the Law of Cosines instead of the Law of Sines in this problem? Why or why not?
4. What is the perimeter of \( \Delta LMN \) using the calculated length of \( l \)?
5. How does the Law of Sines differ from the Law of Cosines in terms of applicability to different triangle problems?

### Tip:
When solving triangle problems, always check if you can simplify your calculations using trigonometric identities or by checking if the triangle is a special type (like isosceles or right-angled).

Find the length of side l in triangle ΔLMN, given side m = 450 inches, ∠M = 74°, and ∠N = 41°, using the Law of Sines and the triangle angle sum theorem. This step-by-step solution is ideal for high school students (Grades 9-12) studying trigonometry.

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