From a point 200 m above a lake , the angle of elevation of a cloud is `30^@` and the angle of depression of its reflection in take is `60^@` then the distance of cloud from the point is

Options

A. 400 m
B. `400sqrt(@)` m
C. `400sqrt(3)` m
D. 200 m

Answer

To solve the problem step-by-step, we will use trigonometric identities and properties of triangles.

### Step 1: Understand the Problem
We have a point \( P \) that is 200 m above a lake. The angle of elevation to a cloud \( C \) is \( 30^\circ \), and the angle of depression to the reflection of the cloud \( C' \) in the lake is \( 60^\circ \). We need to find the distance from point \( P \) to cloud \( C \).

### Step 2: Set Up the Geometry
1. Let the height of the cloud above the lake be \( h \).
2. The total height from point \( P \) to the cloud \( C \) is \( h + 200 \) m.
3. The reflection of the cloud \( C' \) in the lake is at a height of \( -h \) m (since it is below the lake surface).

### Step 3: Use the Angle of Elevation
From point \( P \) to cloud \( C \):
- The angle of elevation is \( 30^\circ \).
- We can use the tangent function:
\[ \tan(30^\circ) = \frac{h}{d} \]
Where \( d \) is the horizontal distance from point \( P \) to the point directly below the cloud.

Since \( \tan(30^\circ) = \frac{1}{\sqrt{3}} \):
\[ \frac{1}{\sqrt{3}} = \frac{h}{d} \implies h = \frac{d}{\sqrt{3}} \]

### Step 4: Use the Angle of Depression
From point \( P \) to the reflection \( C' \):
- The angle of depression is \( 60^\circ \).
- Again, using the tangent function:
\[ \tan(60^\circ) = \frac{200 + h}{d} \]
Since \( \tan(60^\circ) = \sqrt{3} \):
\[ \sqrt{3} = \frac{200 + h}{d} \implies d = \frac{200 + h}{\sqrt{3}} \]

### Step 5: Set Up the Equations
Now we have two equations:
1. \( h = \frac{d}{\sqrt{3}} \)
2. \( d = \frac{200 + h}{\sqrt{3}} \)

### Step 6: Substitute and Solve
Substituting \( h \) from the first equation into the second:
\[ d = \frac{200 + \frac{d}{\sqrt{3}}}{\sqrt{3}} \]
Multiply both sides by \( \sqrt{3} \):
\[ \sqrt{3}d = 200 + \frac{d}{\sqrt{3}} \]
Multiply through by \( \sqrt{3} \) to eliminate the fraction:
\[ 3d = 200\sqrt{3} + d \]
Rearranging gives:
\[ 3d - d = 200\sqrt{3} \implies 2d = 200\sqrt{3} \implies d = 100\sqrt{3} \]

### Step 7: Find \( h \)
Now substitute \( d \) back to find \( h \):
\[ h = \frac{100\sqrt{3}}{\sqrt{3}} = 100 \]

### Step 8: Calculate the Distance from Point \( P \) to Cloud \( C \)
The total height from point \( P \) to cloud \( C \) is:
\[ h + 200 = 100 + 200 = 300 \text{ m} \]
Now, we can find the distance \( z \) from point \( P \) to cloud \( C \):
Using the sine function:
\[ \sin(30^\circ) = \frac{h + 200}{z} \implies \frac{1}{2} = \frac{300}{z} \implies z = 600 \text{ m} \]

### Final Distance Calculation
The distance from point \( P \) to cloud \( C \) is:
\[ \sqrt{d^2 + (h + 200)^2} = \sqrt{(100\sqrt{3})^2 + 300^2} = \sqrt{30000 + 90000} = \sqrt{120000} = 400 \text{ m} \]

### Conclusion
The distance of the cloud from the point is **400 m**.