Assertion (A): The angle of elevation of the top of the tower from a point on the ground, which is 30 m away from the foot of the tower, is `30^(@)`. The height of the tower is 10 m.
Reason (R): The angle of depression from B to A and Angle of elevation from A to B are equal.

Options

A. Both A and R are true and R is the correct explanation of A.
B. Both A and R are true and R is not correct explanation of A.
C. A is true but R is false.
D. A is false but R is true.

Answer

To solve the given problem, we need to analyze the assertion and the reason provided.

### Step 1: Understanding the Assertion
The assertion states that the angle of elevation of the top of the tower from a point on the ground, which is 30 m away from the foot of the tower, is \(30^\circ\) and that the height of the tower is 10 m.

### Step 2: Visualizing the Situation
Let’s denote:
- Point C as the foot of the tower.
- Point B as the top of the tower.
- Point A as the point on the ground 30 m away from C.

We have a right triangle formed by points A, C, and B, where:
- AC = 30 m (distance from the foot of the tower to point A)
- BC = h (height of the tower)
- Angle CAB = \(30^\circ\) (angle of elevation)

### Step 3: Applying Trigonometry
Using the tangent function for the angle of elevation:
\[ \tan(30^\circ) = \frac{BC}{AC} \]
Substituting the known values:
\[ \tan(30^\circ) = \frac{h}{30} \]
We know that \(\tan(30^\circ) = \frac{1}{\sqrt{3}}\), so:
\[ \frac{1}{\sqrt{3}} = \frac{h}{30} \]

### Step 4: Solving for h
Cross-multiplying gives:
\[ h = 30 \cdot \frac{1}{\sqrt{3}} = \frac{30}{\sqrt{3}} = 10\sqrt{3} \text{ m} \]
Calculating \(10\sqrt{3}\):
\[ 10\sqrt{3} \approx 10 \times 1.732 = 17.32 \text{ m} \]

### Step 5: Conclusion for Assertion
The height of the tower is approximately 17.32 m, which contradicts the assertion that the height is 10 m. Therefore, the assertion is **false**.

### Step 6: Analyzing the Reason
The reason states that the angle of depression from point B to point A and the angle of elevation from point A to point B are equal. This is a true statement in trigonometry because the angle of depression and angle of elevation between two points are equal when measured from a horizontal line.

### Final Conclusion
- Assertion (A) is **false**.
- Reason (R) is **true**.