A pole 25 meters long stands on the top of a tower 225 meters high. It `theta` is the angle subtended by the pole at a point on the ground which is at distance 2.25 km, from the foot of the tower, then `tan theta` is equal to

Options

A. `1/90`
B. `1/91`
C. `1/10`
D. `1/9`

Answer

To solve the problem, we need to find the value of \( \tan \theta \), where \( \theta \) is the angle subtended by a pole at a point on the ground. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the Setup
We have a tower that is 225 meters high with a pole of 25 meters on top of it. The total height from the ground to the top of the pole is:
\[ \text{Total Height} = \text{Height of Tower} + \text{Height of Pole} = 225 \, \text{m} + 25 \, \text{m} = 250 \, \text{m} \]

### Step 2: Convert Distance to Meters
The distance from the foot of the tower to the point on the ground is given as 2.25 km. We need to convert this distance into meters:
\[ \text{Distance} = 2.25 \, \text{km} \times 1000 \, \text{m/km} = 2250 \, \text{m} \]

### Step 3: Set Up the Right Triangle
We can visualize a right triangle where:
- The height of the triangle (perpendicular) is the total height of the pole and tower, which is 250 m.
- The base of the triangle (adjacent) is the distance from the foot of the tower to the point on the ground, which is 2250 m.

### Step 4: Calculate \( \tan \theta \)
The tangent of angle \( \theta \) is defined as the ratio of the opposite side to the adjacent side:
\[ \tan \theta = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{\text{Total Height}}{\text{Distance}} = \frac{250 \, \text{m}}{2250 \, \text{m}} \]

### Step 5: Simplify the Expression
Now, we simplify the fraction:
\[ \tan \theta = \frac{250}{2250} = \frac{25}{225} = \frac{1}{9} \]

### Conclusion
Thus, the value of \( \tan \theta \) is:
\[ \tan \theta = \frac{1}{9} \]