Final answer:
To find the distance and height, we calculate using the angles of depression: 45° implies the distance is 50 m, and for 30°, tan(30°) = 50 / (50 + h) gives the height as 50(√3 - 1) m. The correct answers are (i) 50 m for the distance and (ii) 50(√3 - 1) m for the height of the pole, making option (c) the correct choice.
Step-by-step explanation:
The problem you’ve given involves the application of trigonometric concepts to solve for the distance between a tower and a pole and the height of the pole based on given angles of depression. Let’s denote the distance from the tower to the pole as x, and the height of the pole as h. Using the angles of depression and the height of the tower, we can set up right triangles and use basic trigonometry to find the answers.
- For the angle of 45°, we can use the fact that in a 45°-45°-90° triangle, the legs are equal. Since the tower's height is 50 m, the distance from the tower to the bottom of the pole is also 50 m. This means that x = 50 m.
- For the angle of 30°, we use the tangent function, which in a right triangle relates the opposite side (the tower's height) to the adjacent side (the horizontal distance to the top of the pole). The tangent of 30° is 1/√3, so we have tan(30°) = 50 / (x + h), giving us h = 50√3 - 50.
Now, we can solve for h using the value for x we found earlier. Since the top of the pole is 50 m horizontally from the tower (at the 45° depression angle), we can set up an equation: tan(30°) = 50 / (50 + h). Solving for h gives us h = 50√3 - 50. Thus, the height of the pole is 50(√3 - 1) meters.
Finally, by combining the two parts of this problem, we find the answers are: (i) 50 meters and (ii) 50(√3 - 1) meters. Therefore, the correct option is (c).