The pilot of an aircraft flying horizontally at a speed of 1200 Km/hr observes an angle of depression of a point on the ground changes from 30 degree to 45 degree in 15 seconds. Find the height of the aircraft at which it is flying?

at 1200km/hr, the plane travels

1200km/hr * 15sec * hr/3600sec = 5km

If the height of the plane is h, and the horizontal distance of the plane from the point on the ground after the 2nd observation is x,

h/x = tan 45° = 1
h/(x+5) = tan 30° = 1/√3

h=x, so,
h/(h+5) = 1/√3
h = 5/(√3-1) = 6.83 km

To find the height of the aircraft, we can use the trigonometric concept of tangent. The tangent of an angle is defined as the ratio of the length of the opposite side to the length of the adjacent side.

In this case, the angle of depression is changing from 30 degrees to 45 degrees. Let's consider the initial angle of depression as 30 degrees, the final angle of depression as 45 degrees, and the time it takes for the angle to change as 15 seconds.

Let's break down the problem step by step:

Step 1: Convert the speed of the aircraft from km/hr to m/s.
Given, the speed of the aircraft is 1200 km/hr.
To convert km/hr to m/s, divide the speed by 3.6.
1200 km/hr ÷ 3.6 = 333.33 m/s (approx.)

Step 2: Determine the horizontal distance covered by the aircraft.
Since the aircraft is flying horizontally, the horizontal distance covered by the aircraft will be equal to the speed multiplied by the time taken for the angle to change.
Distance = Speed × Time
Distance = 333.33 m/s × 15 s = 5000 m

Step 3: Calculate the height of the aircraft.
To find the height, we need to consider the triangle formed between the aircraft, the point on the ground, and the line of sight from the pilot's perspective.

Let h be the height of the aircraft.
Now, we can consider the tangent of the initial angle of depression and the final angle of depression.
tan(30°) = h / 5000 (using the initial angle of depression)
tan(45°) = h / 5000 (using the final angle of depression)

Step 4: Solve for h.
Rearranging the equations, we have:
h = tan(30°) × 5000
h = tan(45°) × 5000

Using a scientific calculator, calculate the values of tan(30°) and tan(45°):
h ≈ 2886.75 m (approx.)

Therefore, the height of the aircraft at which it is flying is approximately 2886.75 m.

6.83

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