An aeroplane when flying at a height of 4000 m from the ground passes vertically above another plane at an instant of an angle of the two plane from the same point on the ground are 60 degree and 45 degree respectively. Find the vertical distance between the aeroplanes at that instant??????

To find the vertical distance between the two airplanes, we can use some basic trigonometry. We know the height of the first airplane and the angles formed with the ground from a common point. Let's break this down step by step.

Understanding the Problem

We have two airplanes: one flying at a height of 4000 meters and another one whose height we need to determine. The angles formed with the ground from a common point are 60 degrees for the first airplane and 45 degrees for the second airplane.

Using Trigonometry

We can visualize this situation by imagining a right triangle for each airplane. The height of the first airplane is given, and we need to find the height of the second airplane using the angle of elevation.

• The first airplane (A) is at a height of 4000 m.
• The angle of elevation to the first airplane from the ground point (P) is 60 degrees.
• The angle of elevation to the second airplane (B) is 45 degrees.

Calculating the Height of the Second Airplane

For the second airplane, we can use the tangent function, which relates the angle of elevation to the opposite side (height of the airplane) and the adjacent side (distance from the point on the ground to the point directly below the airplane).

The formula for tangent is:

tan(θ) = opposite / adjacent

For the second airplane at 45 degrees:

tan(45°) = height of B / distance from P to B

Since tan(45°) = 1, we can say:

height of B = distance from P to B

Now, for the first airplane at 60 degrees:

tan(60°) = height of A / distance from P to A

Since tan(60°) = √3, we can express this as:

√3 = 4000 / distance from P to A

From this, we can find the distance from point P to airplane A:

distance from P to A = 4000 / √3

Finding the Vertical Distance

Now, we can find the height of the second airplane (B) using the distance we just calculated. Since the distance from P to A is equal to the height of B (because of the 45-degree angle), we have:

height of B = 4000 / √3

Now, we can find the vertical distance between the two airplanes:

Vertical distance = height of A - height of B

Vertical distance = 4000 - (4000 / √3)

To simplify this, we can factor out 4000:

Vertical distance = 4000(1 - 1/√3)

To get a numerical value, we can calculate:

1/√3 is approximately 0.577, so:

1 - 1/√3 ≈ 1 - 0.577 = 0.423

Thus, the vertical distance is:

Vertical distance ≈ 4000 * 0.423 ≈ 1692 m

Final Result

The vertical distance between the two airplanes at that instant is approximately 1692 meters.