A plane is going in circles around an airport. The plane takes 3 minutes to complete one round. The angle of elevation of the plane from a point P on the ground at time t seconds is equal to that at time (t + 30) seconds. At time (t + x) seconds, the plane flies vertically above the point P. What is x equal to?a)75 secondsb)90 secondsc)105 secondsd)135 secondsCorrect answer is option 'C'. Can you explain this answer?

Question

Answer

Let the plane be at point A at t seconds and at point B after t + 30 seconds
Since the motion is uniform, we can say that at time t+15 seconds, the plane is above the point is diametrically opposite to the point P from where the angle is same.
Now since the time taken to cover the full circle is 3 minutes (180 seconds), the time taken by the plane to reach the diametrically opposite point will be 90 seconds.
So the time after which the plane reaches the point P will be = t+ 15 + 90 seconds = (t + 105 )seconds

Answer

Let the plane be at point A at t seconds and at point B after t + 30 seconds
Since the motion is uniform, we can say that at time t+15 seconds, the plane is above the point is diametrically opposite to the point P from where the angle is same.
Now since the time taken to cover the full circle is 3 minutes (180 seconds), the time taken by the plane to reach the diametrically opposite point will be 90 seconds.
So the time after which the plane reaches the point P will be = t+ 15 + 90 seconds = (t + 105 )seconds

Answer

Problem Analysis

The plane is flying in circles around an airport, completing one round in 3 minutes. We need to find the value of x, which represents the time in seconds when the plane is directly above point P.

Solution

Let's break down the problem step by step:

Step 1: Determine the angle of elevation at time t seconds.
Since the plane takes 3 minutes to complete one round, it covers 360 degrees in 3 minutes. Therefore, the plane covers 360/3 = 120 degrees per minute.
Let's assume the angle of elevation at time t seconds is θ degrees. Then, at time (t + 30) seconds, the plane will have covered an additional 120/60 × 30 = 60 degrees.
Thus, the angle of elevation at time (t + 30) seconds is θ + 60 degrees.

Step 2: Find the value of θ.
Since the angle of elevation at time t seconds is equal to that at time (t + 30) seconds, we can write the equation:
θ = θ + 60
Subtracting θ from both sides, we get:
0 = 60
This equation is not possible, indicating that our assumption was incorrect.
Therefore, there must be a mistake in the problem statement or conditions.

Conclusion: The given problem has no valid solution based on the provided conditions. It is important to carefully analyze the given information and conditions to ensure that a valid solution can be obtained.

Answer

Let the plane be at point A at t seconds and at point B after t + 30 seconds
Since the motion is uniform, we can say that at time t+15 seconds, the plane is above the point is diametrically opposite to the point P from where the angle is same.
Now since the time taken to cover the full circle is 3 minutes (180 seconds), the time taken by the plane to reach the diametrically opposite point will be 90 seconds.
So the time after which the plane reaches the point P will be = t+ 15 + 90 seconds = (t + 105 )seconds

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