A boat is 375 teet from the base of a cillt. Andrew, whe is at the top of the eillt, noticea that the angle ef depreasien to the boat is 64 . How high is the cilfre Round to the nearest tenth. 642.7 teet 417.2 teet 182.9 teet 768.9 teet

Sure, let's solve the problem step by step. ### **Problem Statement:** A boat is **375 feet** from the base of a **cliff**. Andrew, who is at the top of the **cliff**, notices that the **angle of depression** to the boat is **64 degrees**. How high is the cliff? **Round to the nearest tenth**. **Options:** - 642.7 feet - 417.2 feet - 182.9 feet - 768.9 feet ### **Understanding the Problem:** - **Angle of Depression:** The angle formed by the line of sight when looking downward from a horizontal line. - **Given:** - Distance from the boat to the base of the cliff (horizontal distance): **375 feet** - Angle of depression: **64°** - **Find:** Height of the cliff (**h**) ### **Diagram:** Imagine a right triangle where: - The **horizontal side** (adjacent to the angle) is **375 feet**. - The **vertical side** (opposite to the angle) is the height of the cliff (**h**) we need to find. - The **angle** at the top of the cliff is **64°**. ### **Trigonometric Relationship:** 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: \[ \tan(64°) = \frac{h}{375} \] ### **Solving for Height (h):** 1. **Calculate \(\tan(64°)\):** \[ \tan(64°) \approx 2.0503 \] 2. **Multiply both sides by 375 to solve for \(h\):** \[ h = 375 \times \tan(64°) \\ h \approx 375 \times 2.0503 \\ h \approx 768.8625 \text{ feet} \] 3. **Round to the nearest tenth:** \[ h \approx 768.9 \text{ feet} \] ### **Answer:** **768.9 feet**