is looking up at the cliff of a mountain. He is 50 m from I 0 and the angle of elevation from where he stands is \( 66^{\circ} \). icular height of the mountain. (Use question 4 as a guidelit ram is not drawn according to scale)

It appears you're working on a trigonometry problem involving the angle of elevation to determine the height of a mountain. Let's break down the problem and solve it step-by-step. ### **Problem Statement:** A person is standing **50 meters** away from the base of a mountain. The angle of elevation from the person's position to the top of the mountain is **\(66^{\circ}\)**. Determine the height of the mountain. ### **Diagram:** While a diagram isn't drawn to scale, envision the scenario as a right-angled triangle where: - The **horizontal distance** from the person to the base of the mountain is **50 meters**. - The **angle of elevation** from the person's eye to the top of the mountain is **\(66^{\circ}\)**. - The **vertical side** represents the **height of the mountain (h)** that we need to find. ``` | |\ | \ | \ | \ |____\ 50m 66° ``` ### **Solution:** 1. **Identify the Right Triangle Components:** - **Adjacent side (base):** 50 meters - **Opposite side (height of the mountain):** \( h \) meters - **Angle of elevation:** \(66^{\circ}\) 2. **Use the Tangent Function:** The tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side. \[ \tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} \] Plugging in the known values: \[ \tan(66^{\circ}) = \frac{h}{50} \] 3. **Solve for \( h \):** \[ h = 50 \times \tan(66^{\circ}) \] 4. **Calculate \( \tan(66^{\circ}) \):** Using a calculator: \[ \tan(66^{\circ}) \approx 2.2460 \] 5. **Determine the Height \( h \):** \[ h = 50 \times 2.2460 = 112.3 \text{ meters} \] Rounding to a reasonable degree of precision: \[ h \approx 112 \text{ meters} \] ### **Conclusion:** The height of the mountain is approximately **112 meters**.