opic 6: Angle of Elevation \& Angle of Depression irecfions: Use a picture to help solve the problem. Round your answer to the nearest tenth. 5. The town park does an outdoor movie night every Saturday during the summer on a large screen. Kate is sitting 36 feet from the base of the screen, watching a movie with her family. the angle of elevation from Kate to the top of the screen is \( 24^{\circ} \), how tall is the movie screen

To determine the height of the movie screen using the angle of elevation, follow these steps: ### **1. Visualize the Problem with a Right Triangle** Imagine a right triangle where: - **Adjacent Side (Base):** The horizontal distance from Kate to the base of the screen, which is **36 feet**. - **Opposite Side (Height):** The height of the movie screen that we need to find. - **Angle of Elevation:** The angle between the ground and Kate's line of sight to the top of the screen, which is **\(24^{\circ}\)**. ``` /| / | / | / | Height (h) / | /_____| 36 ft Angle = 24° ``` ### **2. Use Trigonometric Ratios** In a right triangle, the **tangent** of an angle is the ratio of the opposite side to the adjacent side. Mathematically, this is expressed as: \[ \tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} \] Where: - \(\theta = 24^{\circ}\) - Opposite = Height of the screen (\(h\)) - Adjacent = 36 feet ### **3. Set Up the Equation** \[ \tan(24^{\circ}) = \frac{h}{36} \] ### **4. Solve for the Height (\(h\))** \[ h = 36 \times \tan(24^{\circ}) \] ### **5. Calculate Using a Calculator** First, find \(\tan(24^{\circ})\): \[ \tan(24^{\circ}) \approx 0.4452 \] Then, multiply by 36: \[ h = 36 \times 0.4452 \approx 16.0272 \text{ feet} \] ### **6. Round to the Nearest Tenth** \[ h \approx 16.0 \text{ feet} \] ### **Final Answer** The movie screen is approximately **16.0 feet** tall.