There is a 7m high statue standing on a cliff. At a point P on the ground, the angle of elevation of the foot of the statue is `alpha`. After walking 34 metres towords the cliff from that point, the angle of elevation of the top of the statue is `(90^(@)-alpha)`, find the height of the cliff if `tanalpha=1/2`.

Answer

To solve the problem step by step, we will use trigonometric relationships based on the information provided in the question.

### Step 1: Understand the scenario
We have a statue that is 7 meters high standing on top of a cliff. We need to find the height of the cliff (let's denote it as \( h \)). The angle of elevation to the foot of the statue from point \( P \) is \( \alpha \), and after walking 34 meters towards the cliff, the angle of elevation to the top of the statue becomes \( 90^\circ - \alpha \).

### Step 2: Set up the relationships
From point \( P \):
- The height of the cliff is \( h \).
- The height of the statue is 7 meters.
- The total height from the ground to the top of the statue is \( h + 7 \).

### Step 3: Use the tangent function for angle \( \alpha \)
From point \( P \), we can write:
\[ \tan \alpha = \frac{h}{x} \]
where \( x \) is the horizontal distance from point \( P \) to the base of the cliff. Given that \( \tan \alpha = \frac{1}{2} \), we have:
\[ \frac{1}{2} = \frac{h}{x} \implies x = 2h \]

### Step 4: Use the tangent function for angle \( 90^\circ - \alpha \)
After walking 34 meters towards the cliff, the new distance to the base of the cliff becomes \( x - 34 \). The angle of elevation to the top of the statue is \( 90^\circ - \alpha \), so we can write:
\[ \tan(90^\circ - \alpha) = \cot \alpha = \frac{1}{\tan \alpha} = 2 \]
Thus, we can write:
\[ 2 = \frac{h + 7}{x - 34} \]

### Step 5: Substitute \( x \) in the equation
Substituting \( x = 2h \) into the equation gives:
\[ 2 = \frac{h + 7}{2h - 34} \]

### Step 6: Cross-multiply to solve for \( h \)
Cross-multiplying yields:
\[ 2(2h - 34) = h + 7 \]
Expanding this gives:
\[ 4h - 68 = h + 7 \]

### Step 7: Rearranging the equation
Rearranging the equation to isolate \( h \):
\[ 4h - h = 7 + 68 \]
\[ 3h = 75 \]

### Step 8: Solve for \( h \)
Dividing both sides by 3:
\[ h = \frac{75}{3} = 25 \]

### Conclusion
The height of the cliff is \( 25 \) meters.