Question: An awning that covers a sliding glass door that is $88$ inches tall forms an angle of $50^\circ$ with the wall. The purpose of the awning is to prevent sunlight from entering the house when the angle of elevation of the sun is more than $X = 80^\circ$. See the figure. Find the length $L$ of the awning.
1. **State the problem:** We need to find the length $L$ of an awning that forms a $50^\circ$ angle with a vertical wall that is $88$ inches tall.
2. **Understand the triangle:** The awning, wall, and ground form a right triangle. The wall is vertical with length $88$ inches, the awning is the hypotenuse $L$, and the angle between the awning and the wall is $50^\circ$.
3. **Identify the trigonometric relation:** Since the wall is adjacent to the angle $50^\circ$ and the awning is the hypotenuse, we use the cosine function:
$$\cos(50^\circ) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{88}{L}$$
4. **Solve for $L$:**
$$L = \frac{88}{\cos(50^\circ)}$$
5. **Calculate $\cos(50^\circ)$:**
$$\cos(50^\circ) \approx 0.6428$$
6. **Substitute and compute:**
$$L = \frac{88}{0.6428} \approx 136.9$$
7. **Interpretation:** The length of the awning $L$ is approximately $136.9$ inches.
**Final answer:**
$$L \approx 136.9 \text{ inches}$$