The problem involves solving for the length of $l$ in $\triangle LMN$ using the given information:

• $m = 45$ inches (side length),
• $m\angle M = 74^\circ$,
• $m\angle N = 41^\circ$.

To solve this, we will use the Law of Sines:

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

Step 1: Find $\angle L$

The angles of a triangle sum to $180^\circ$. So:

$m\angle L = 180^\circ - m\angle M - m\angle N = 180^\circ - 74^\circ - 41^\circ = 65^\circ.$

Step 2: Apply the Law of Sines

The triangle has:

• Side $m = 45$ inches opposite $\angle L = 65^\circ$,
• Side $l$ opposite $\angle N = 41^\circ$.

Using the Law of Sines:

$\frac{l}{\sin 41^\circ} = \frac{45}{\sin 65^\circ}.$

Rearranging for $l$:

$l = \frac{45 \cdot \sin 41^\circ}{\sin 65^\circ}.$

Step 3: Calculate $l$

Using a calculator:

• $\sin 41^\circ \approx 0.6561$,
• $\sin 65^\circ \approx 0.9063$.

Substitute these values:

$l = \frac{45 \cdot 0.6561}{0.9063} \approx \frac{29.5245}{0.9063} \approx 32.6 \, \text{inches}.$

Final Answer:

The length of $l$ is approximately 32.6 inches.

Would you like to see additional steps or explanations? Here are some related questions for practice:

1. How does the Law of Sines work for obtuse triangles?
2. How can the Law of Cosines be used in place of the Law of Sines?
3. What happens if the triangle is a right triangle? How do the trigonometric laws simplify?
4. Can you derive the Law of Sines from a geometric perspective?
5. How does rounding affect the accuracy of trigonometric solutions?

Tip: Always recheck the angle sum in triangle problems to ensure calculations are consistent.