A temple and a flagstaff surmounted at its top, each subtends equal angle of `30^(@)` at a point on the ground. If the height of the temple is 10 m, find the height of the flagstaff.

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the problem
We have a temple of height 10 m and a flagstaff on top of it. Both the temple and the flagstaff subtend an angle of 30° at a point on the ground. We need to find the height of the flagstaff.

### Step 2: Set up the diagram
Let:
- The height of the temple = 10 m
- The height of the flagstaff = H m
- The horizontal distance from the point on the ground to the base of the temple = D m

### Step 3: Analyze the triangle formed by the temple
In the triangle formed by the temple, we can use the tangent function:
- The angle subtended by the temple at the point on the ground is 30°.
- Therefore, we can write:
\[ \tan(30°) = \frac{\text{Height of the temple}}{\text{Distance from the point to the temple}} = \frac{10}{D} \]

### Step 4: Calculate D using tan(30°)
We know that:
\[ \tan(30°) = \frac{1}{\sqrt{3}} \]
So, we can set up the equation:
\[ \frac{1}{\sqrt{3}} = \frac{10}{D} \]
Cross-multiplying gives us:
\[ D = 10\sqrt{3} \text{ m} \]

### Step 5: Analyze the triangle formed by the flagstaff
Now, we will consider the entire height (temple + flagstaff) and the angle subtended:
- The total height from the ground to the top of the flagstaff is \(10 + H\).
- The angle subtended at the point on the ground is now 60° (since both angles are equal).

Using the tangent function again:
\[ \tan(60°) = \frac{\text{Total height}}{\text{Distance from the point to the base of the temple}} = \frac{10 + H}{D} \]

### Step 6: Calculate using tan(60°)
We know that:
\[ \tan(60°) = \sqrt{3} \]
So, we can set up the equation:
\[ \sqrt{3} = \frac{10 + H}{10\sqrt{3}} \]
Cross-multiplying gives us:
\[ 10\sqrt{3} \cdot \sqrt{3} = 10 + H \]
This simplifies to:
\[ 30 = 10 + H \]

### Step 7: Solve for H
Now, we can solve for H:
\[ H = 30 - 10 = 20 \text{ m} \]

### Final Answer
The height of the flagstaff is **20 meters**.

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