1. **State the problem:** We need to find the length $x$ of a retractable awning that is attached to a wall at a height of $y=12$ feet and lowers at an angle of $50^\circ$ from the vertical wall. The sun's rays form an angle of elevation of $70^\circ$. We want to find $x$ such that no direct sunlight enters the door when the sun's elevation is greater than $70^\circ$. 2. **Understand the triangle:** The awning, the wall, and the ground form a right triangle. The awning is the hypotenuse $x$, the vertical wall is the opposite side with length $y=12$, and the angle between the awning and the wall is $50^\circ$. 3. **Use trigonometry:** Since $x$ is the hypotenuse and $y$ is the side adjacent to the angle $50^\circ$ (because the angle is between awning and wall), we use the cosine function: $$\cos(50^\circ) = \frac{y}{x}$$ 4. **Solve for $x$:** $$x = \frac{y}{\cos(50^\circ)}$$ 5. **Calculate $x$:** $$x = \frac{12}{\cos(50^\circ)}$$ Calculate $\cos(50^\circ)$: $$\cos(50^\circ) \approx 0.6428$$ So, $$x = \frac{12}{0.6428} \approx 18.67$$ 6. **Final answer:** The length of the awning is approximately **18.67 feet**.