Final answer:
Using trigonometry, we determine that Paul is 153m above the ground by using the tangent of the angle of elevation, which equals the angle of depression due to a straight climb. The third angle, representing the angle of depression from the starting point, is 21.6 degrees.
Step-by-step explanation:
The question is about using trigonometry to solve for the height of a point on a pyramid-shaped hill and determining the size of the third angle of depression from that point. To find the height of Paul above the ground, we can use the tan function of the angle of depression, 68.4 degrees, which gives us the relationship between the opposite side (height from ground) and the adjacent side (the climb of 153m). However, in this case, since the angle is more than 45 degrees, we should use the angle of elevation from the ground to Paul, which would be the complement to 90 degrees: 90 - 68.4 = 21.6 degrees. Therefore, the height (h) can be calculated using the tangent of 21.6 degrees:
- tan(21.6 degrees) = h / 153m
- h = tan(21.6 degrees) × 153m
- h = 153m (since when the angle of elevation equals the angle of depression, the height is equal to the horizontal distance climbed)
For the size of the third angle, in this context, it refers to the angle of depression from Paul's starting point. Since Paul's climb is directly upwards, the third angle is the same as the angle of depression, which is 68.4 degrees. Therefore, the correct answer is A: Paul is 153m above the ground; the third angle = 21.6 degrees.