A plane is moving in circles around an airport and takes 3 minutes to complete one full round. The angle of elevation of the plane from a point P on the ground at time t seconds is equal to the angle of elevation at time (t + 30) seconds. At time (t + x) seconds, the plane is flying vertically above point P. What is the value of x (in seconds)?

Introduction / Context:
This is a classic time and distance problem combined with circular motion and symmetry. The plane is flying in a horizontal circle around an airport, and an observer on the ground measures the angle of elevation of the plane at different times. When the angle of elevation is the same at two different instants, the corresponding positions of the plane on the circular path must be symmetric with respect to a particular line. The problem then asks at what later time the plane will be exactly vertically above the observer, which is a position directly related to the symmetry of the circular motion.

Given Data / Assumptions:

The plane takes 3 minutes to complete one round, so the time period T = 3 minutes = 180 seconds.
The angle of elevation from point P at time t seconds equals the angle of elevation at time t + 30 seconds.
At time t + x seconds, the plane is vertically above the observer at P.
The plane moves at constant speed along a circular horizontal path around the airport.

Concept / Approach:
For uniform circular motion, equal angles of elevation as seen from a fixed point correspond to symmetric positions on the circle. Let the positions at times t and t + 30 seconds be B and C respectively. These positions are symmetric about the point O that is diametrically opposite to P on the circle, because from B and C the line of sight from P makes the same angle. This symmetry implies that the plane reaches O at the mid time between t and t + 30, namely t + 15 seconds. Once the plane is at O, it still has to travel half a circle, or 180 degrees of angular distance, to reach directly above P at point A.

Step-by-Step Solution:
Step 1: Compute the time for one full revolution. T = 3 minutes = 180 seconds. Step 2: Use symmetry of equal angles of elevation. If the angle from P is the same at t and t + 30 seconds, the plane is at two symmetric points B and C. The midpoint in time between these events is t + 15 seconds. At time t + 15 seconds, the plane is at the point O, diametrically opposite to P on the circle. Step 3: Time from O to P. From O to P is half of the circle, which takes T / 2 = 180 / 2 = 90 seconds. Step 4: Combine times. Total time from t to the plane being vertically above P is 15 + 90 = 105 seconds. Therefore x = 105 seconds.

Verification / Alternative check:
We can also think in terms of angular distance. Let B and C be symmetric with respect to the diameter through O and P. The plane takes 30 seconds to move from B to C and 180 seconds for a full 360 degree revolution. Thus, the angular spacing between B and C is proportional to 30 seconds, while the path from O to P corresponds to a half revolution or 90 seconds. Since O is the midpoint in time between B and C, we again get 15 seconds from B to O and 90 seconds from O to P, making a total of 105 seconds from t to when the plane is above P.

Why Other Options Are Wrong:
Option a (75 seconds) and option b (90 seconds) ignore either the additional half revolution from O to P or the mid interval of 15 seconds. Option d (135 seconds) overcounts the required time and does not follow from any symmetric splitting of the circle given the 30 second gap. Only 105 seconds satisfies both the symmetry of the angles and the total time needed to cover the remaining half revolution.

Common Pitfalls:
A common mistake is to assume that the plane is directly above P exactly halfway between t and t + 30 seconds, which would give 15 seconds, clearly not in the options. Another error is to forget that the point of symmetry for equal angles of elevation is diametrically opposite P, not directly above P. Some candidates also divide the full time period 180 seconds incorrectly or fail to account for the half revolution between O and P. Carefully identifying the geometry of the circle and the role of symmetry avoids these traps.

Final Answer:
Thus, the plane is vertically above point P after 105 seconds, so x = 105 seconds.