A radio tower is 200 feet tall. How far is an observer from the base of the tower if the angle of their line of sight hrom the ground to the top of the tower is \( 65^{\circ} \) ? Round to the nearest hundredth. (t point)

To determine how far the observer is from the base of the 200-foot-tall radio tower with an angle of elevation of \( 65^{\circ} \), we can use trigonometry, specifically the tangent function. **Step-by-Step Solution:** 1. **Understand the Right Triangle:** - **Opposite Side (Height of the Tower):** 200 feet - **Angle of Elevation:** \( 65^{\circ} \) - **Adjacent Side (Distance from Observer to Base):** \( x \) (what we're trying to find) 2. **Use the Tangent Function:** \[ \tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} \] Plugging in the known values: \[ \tan(65^{\circ}) = \frac{200}{x} \] 3. **Solve for \( x \):** \[ x = \frac{200}{\tan(65^{\circ})} \] 4. **Calculate \( \tan(65^{\circ}) \):** \[ \tan(65^{\circ}) \approx 2.1445 \] 5. **Compute the Distance:** \[ x \approx \frac{200}{2.1445} \approx 93.17 \text{ feet} \] **Final Answer:** The observer is approximately **93.17 feet** from the base of the tower.