To solve the problem of finding the height of the tree based on the information given, we will break down the steps clearly and methodically. ### Given Information: - Height of the man = 1.7 m - Angle of elevation to the bird = 30° - Distance from the man to the bird = 25 m ### Step-by-Step Explanation: 1. **Understanding the Problem**: - The man is looking up at the bird at an angle of 30°. This means we can visualize a right triangle where: - The height of the tree (which we need to find) is the vertical side (opposite the angle). - The horizontal distance from the man to the base of the tree is the adjacent side. - The hypotenuse is the line of sight from the man to the bird. 2. **Using Trigonometry**: - In a right triangle, the tangent of an angle is defined as the ratio of the opposite side to the adjacent side. Here, we can use the tangent function: \[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \] - For our case: \[ \tan(30°) = \frac{\text{height of the tree} - \text{height of the man}}{\text{distance from the man to the base of the tree}} \] 3. **Calculating the Height of the Tree**: - We know that: \[ \tan(30°) = \frac{1}{\sqrt{3}} \approx 0.577 \] - Let \( h \) be the height of the tree. The height of the tree above the man's head is \( h - 1.7 \) m. - The distance from the man to the base of the tree is the hypotenuse (25 m) multiplied by the cosine of the angle: \[ \text{adjacent} = 25 \cdot \cos(30°) = 25 \cdot \frac{\sqrt{3}}{2} \approx 21.65 \text{ m} \] - Now, we can set up the equation: \[ \tan(30°) = \frac{h - 1.7}{21.65} \] - Plugging in the values: \[ \frac{1}{\sqrt{3}} = \frac{h - 1.7}{21.65} \] - Cross-multiplying gives: \[ 21.65 = (h - 1.7) \cdot \sqrt{3} \] - Solving for \( h \): \[ h - 1.7 = \frac{21.65}{\sqrt{3}} \approx 12.5 \] \[ h = 12.5 + 1.7 = 14.2 \text{ m} \] ### Conclusion: The height of the tree is **14.2 m**. Therefore, the correct option is **B**. ### Explanation of Other Options: - **Option A (26.7 m)**: This option is incorrect because it does not take into account the correct application of the tangent function and the distance involved. - **Option C**: This option appears to be a formula that is not correctly structured or relevant to the problem as it does not yield a valid height. - **Option D**: Similar to option C, this option does not provide a valid calculation or relevant formula for the height of the tree. ### Revision Summary: - Use trigonometric functions (like tangent) to relate angles and sides in right triangles. - Remember to account for the height of the observer when calculating the total height. - Always check the calculations step-by-step to avoid common pitfalls in trigonometric problems. - Ensure that the units are consistent throughout the calculations.