Question: An awning that covers a sliding glass door that is 88 inches tall forms an angle of 50\degree 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 $X = 80\degree$. See the figure. Find the length $L$ of the awning.
Door = 88 in
$A = 50\degree$
$L$
Sun
$X$
$L \approx$ __ in
(Do not round until the final answer. Then round to one decimal place as needed.)
1. **State the problem:**
We have a vertical door of height $88$ inches and an awning forming an angle of $50\degree$ with the wall. We want to find the length $L$ of the awning that blocks sunlight when the sun's elevation angle $X$ is more than $80\degree$.
2. **Understand the geometry:**
The awning forms a right triangle with the wall (door) and the awning length $L$ as the hypotenuse. The angle between the awning and the wall is $50\degree$.
3. **Identify the formula:**
Since the door height is opposite the angle $A=50\degree$ in the right triangle, we use the sine function:
$$\sin(A) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{88}{L}$$
4. **Solve for $L$:**
$$L = \frac{88}{\sin(50\degree)}$$
5. **Calculate the sine value:**
$$\sin(50\degree) \approx 0.7660$$
6. **Substitute and compute:**
$$L = \frac{88}{0.7660} \approx 114.9$$
7. **Final answer:**
$$L \approx 114.9 \text{ inches}$$
This length ensures the awning blocks sunlight when the sun's elevation angle is more than $80\degree$.