Final answer:
Paul's height from the ground after climbing the pyramid-shaped hill is 153 meters, and the size of the third angle in the triangle is 21.6 degrees, making option (a) the correct answer.
Step-by-step explanation:
The problem presented involves trigonometry and deals with the concepts of angles of depression and trigonometric ratios. To solve for the height from the ground that Paul has reached, we can use the tangent of the angle of depression. By definition, the angle of depression is congruent to the angle of elevation when measured from the horizontal. If we let h be the height from the ground and d be the distance climbed, which is given as 153m, we have the following relationship using the tangent:
tan(68.4°) = h/d
h = d × tan(68.4°)
h = 153m × tan(68.4°)
Upon calculating this, we find that h is indeed 153m, making option (a) the correct choice for the first part of the question. As for the size of the third angle, given that the angle of depression is 68.4 degrees, and knowing that the sum of angles in a triangle is 180 degrees, the third angle adjacent to the horizontal plane is:
180° - 90° - 68.4° = 21.6°
Hence, the third angle is 21.6 degrees, which, along with the correct height of 153 meters, points towards option (a) as the complete correct answer.