A pole of 50 meter high stands on a building 250 m high. To an observer at a height of 300 m, thebuilding and the pole subtend equal angles. The distance of the observer from the top of The pole

Answer

To solve the problem step by step, let's break it down clearly:

### Given:
- Height of the pole (PQ) = 50 m
- Height of the building (QR) = 250 m
- Height of the observer (O) = 300 m
- The angles subtended by the building and the pole at the observer are equal.

### Objective:
Find the distance (d) from the observer to the base of the pole.

### Step 1: Understand the Geometry
We have:
- A pole of height 50 m on top of a building of height 250 m.
- The total height from the ground to the top of the pole is \( 250 + 50 = 300 \) m.
- The observer is at a height of 300 m.

### Step 2: Set Up the Angles
Let:
- The angle subtended by the pole at the observer be \( \theta \).
- The angle subtended by the building at the observer be \( \theta \) as well.

### Step 3: Use the Tangent Function
For the pole:
- The height of the pole is 50 m.
- The distance from the observer to the base of the pole is \( d \).

Using the tangent function:
\[ \tan(\theta) = \frac{\text{Height of the pole}}{\text{Distance to the pole}} = \frac{50}{d} \]

For the building:
- The height of the building is 250 m.

Using the tangent function:
\[ \tan(2\theta) = \frac{\text{Height of the building}}{\text{Distance to the building}} = \frac{300}{d} \]

### Step 4: Use the Double Angle Formula
We know that:
\[ \tan(2\theta) = \frac{2\tan(\theta)}{1 - \tan^2(\theta)} \]

Substituting \( \tan(\theta) = \frac{50}{d} \):
\[ \tan(2\theta) = \frac{2 \cdot \frac{50}{d}}{1 - \left(\frac{50}{d}\right)^2} \]

### Step 5: Set Up the Equation
Now we can equate the two expressions for \( \tan(2\theta) \):
\[ \frac{2 \cdot \frac{50}{d}}{1 - \left(\frac{50}{d}\right)^2} = \frac{300}{d} \]

### Step 6: Cross Multiply
Cross multiplying gives:
\[ 2 \cdot 50 = 300 \left(1 - \frac{2500}{d^2}\right) \]
\[ 100 = 300 - \frac{750000}{d^2} \]
\[ \frac{750000}{d^2} = 200 \]

### Step 7: Solve for \( d^2 \)
Rearranging gives:
\[ d^2 = \frac{750000}{200} = 3750 \]

### Step 8: Find \( d \)
Taking the square root:
\[ d = \sqrt{3750} = 25\sqrt{6} \]

### Conclusion
The distance from the observer to the base of the pole is \( 25\sqrt{6} \) meters.

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