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An awning that covers a sliding glass door that is 8888 inches tall forms an angle of 5050 degrees 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=55X = 55 degrees. See the figure. Find the length LL of the awning.

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Full solution

Q. An awning that covers a sliding glass door that is 8888 inches tall forms an angle of 5050 degrees 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=55X = 55 degrees. See the figure. Find the length LL of the awning.
  1. Identify Triangle: Identify the triangle formed by the awning, wall, and ground. The awning forms the hypotenuse, and the height of the door (8888 inches) is the opposite side of the 5050-degree angle.
  2. Use Sine Function: Use the sine function to find the length of the awning (hypotenuse). The sine of an angle in a right triangle is the opposite side divided by the hypotenuse. So, sin⁡(50∘)=88L\sin(50^\circ) = \frac{88}{L}.
  3. Rearrange Equation: Rearrange the equation to solve for LL: L=88sin⁡(50°)L = \frac{88}{\sin(50°)}. Calculate sin⁡(50°)\sin(50°) using a calculator.
  4. Perform Calculation: Perform the calculation: sin⁡(50∘)≈0.7660\sin(50^\circ) \approx 0.7660. So, L=880.7660≈114.88L = \frac{88}{0.7660} \approx 114.88 inches.

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