An aeroplane flying at a height of 300 m above the ground passes vertically above another plane at an instant when the angles of elevation of two planes from the same point on the ground are `60^(@) and 45^(@),` respectively. What is the height of the lower plane from the ground?

Options

A. 50 m
B. `(100)/(sqrt3) m `
C. `100 sqrt3 m `
D. ` 150 ( sqrt3 +1 ) m`

Answer

To solve the problem step by step, we will use trigonometric principles related to angles of elevation.

### Step-by-Step Solution:

1. **Understand the Problem:**
- We have two planes: Plane A flying at a height of 300 m and Plane B at an unknown height (h).
- The angles of elevation from a point on the ground to Plane A and Plane B are 60° and 45°, respectively.

2. **Define the Variables:**
- Let the point on the ground be point O.
- Let the height of Plane B from the ground be h meters.
- The height of Plane A is given as 300 m.

3. **Using Trigonometry for Plane A:**
- From point O, the angle of elevation to Plane A is 60°.
- We can use the tangent function:
\[ \tan(60°) = \frac{\text{Height of Plane A}}{\text{Distance from point O to the point directly below Plane A}} \]
- Let the distance from point O to the point directly below Plane A be \( x \).
- Therefore, we have:
\[ \tan(60°) = \frac{300}{x} \]
- Since \( \tan(60°) = \sqrt{3} \), we can write:
\[ \sqrt{3} = \frac{300}{x} \]
- Rearranging gives:
\[ x = \frac{300}{\sqrt{3}} = 100\sqrt{3} \text{ meters} \]

4. **Using Trigonometry for Plane B:**
- From point O, the angle of elevation to Plane B is 45°.
- Again using the tangent function:
\[ \tan(45°) = \frac{\text{Height of Plane B}}{\text{Distance from point O to the point directly below Plane B}} \]
- Since the distance from point O to the point directly below Plane B is also \( x \), we have:
\[ \tan(45°) = \frac{h}{x} \]
- Since \( \tan(45°) = 1 \), we can write:
\[ 1 = \frac{h}{x} \]
- Rearranging gives:
\[ h = x \]
- Substituting the value of \( x \):
\[ h = 100\sqrt{3} \text{ meters} \]

5. **Conclusion:**
- The height of the lower plane (Plane B) from the ground is \( 100\sqrt{3} \) meters.

### Final Answer:
The height of the lower plane from the ground is \( 100\sqrt{3} \) meters.