Thread: Awning Design

A retractable awning lowers at an angle of 50 degrees from the top of a patio door that is 7 feet above the ground. The awning forms a right triangle. The hypotenuse is the length x of the awning. Find x if no direct sunlight is to enter the door when the angle of elevation of the sun is greater than 70 degrees.

Start by drawing a picture! Draw a vertical line, representing the side of the house. At the top of the line, draw an line going down at 50 degrees from the horizontal, representing the awning. Label the length of the vertical line "7" for the 7 foot height. Label the length of the other line "x" since that length is what you want to find. Finally draw a line up from the base of the "door" (the vertical line) to the end of the "awning". That line represents a ray of sun light that just misses going in the door. Since you want "no direct sunlight to enter the door when the angle of elevation of the sun is greater than 70 degrees", the angle that line makes with the horizontal is 70 degrees. You now have a triangle with one side of length 7 and one side of unknown length "x". The 50 and 70 degree angles are NOT in that triangle because they are with the horizontal, not the vertical. But it is easy to see that the corresponding angles inside the triangle are 90- 70= 20 degrees and 90- 50= 40 degrees. The third angle of the triangle, opposite the 7 foot side, is 180- 20- 40= 120 degrees. Now you can use the sine law to find x.

Originally Posted by HallsofIvy

Start by drawing a picture! Draw a vertical line, representing the side of the house. At the top of the line, draw an line going down at 50 degrees from the horizontal, representing the awning. Label the length of the vertical line "7" for the 7 foot height. Label the length of the other line "x" since that length is what you want to find. Finally draw a line up from the base of the "door" (the vertical line) to the end of the "awning". That line represents a ray of sun light that just misses going in the door. Since you want "no direct sunlight to enter the door when the angle of elevation of the sun is greater than 70 degrees", the angle that line makes with the horizontal is 70 degrees. You now have a triangle with one side of length 7 and one side of unknown length "x". The 50 and 70 degree angles are NOT in that triangle because they are with the horizontal, not the vertical. But it is easy to see that the corresponding angles inside the triangle are 90- 70= 20 degrees and 90- 50= 40 degrees. The third angle of the triangle, opposite the 7 foot side, is 180- 20- 40= 120 degrees. Now you can use the sine law to find x.

Wonderfully done!