What is the angles subtended by 1 m pole at a distance 1 km on the ground in sexagesimal measure ?

Options

A. `(9)/(50pi)` degree
B. `(9)/(5pi)`degree
C. 3.4 minute
D. 3.5 minute

Explanation

To find the angle subtended by a 1 m pole at a distance of 1 km on the ground, we can use trigonometry. Here’s a step-by-step solution:

### Step 1: Understand the Problem
We have a pole of height 1 meter (AB) and a distance of 1 kilometer (BC) from the base of the pole to the point where we want to measure the angle (C).

### Step 2: Convert the Distance
Convert the distance from kilometers to meters for consistency:
1 km = 1000 meters.
So, BC = 1000 meters.

### Step 3: Draw the Diagram
Draw a right triangle where:
- AB is the height of the pole (1 meter).
- BC is the distance from the base of the pole to the point on the ground (1000 meters).
- AC is the line of sight from point C to the top of the pole.

### Step 4: Identify the Angle
Let θ be the angle subtended at point C by the pole. We need to find θ.

### Step 5: Use the Tangent Function
Using the definition of the tangent function in a right triangle:
\[ \tan(θ) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{AB}{BC} \]
Substituting the known values:
\[ \tan(θ) = \frac{1}{1000} \]

### Step 6: Find the Angle θ
To find θ, we take the arctangent (inverse tangent) of both sides:
\[ θ = \tan^{-1}\left(\frac{1}{1000}\right) \]

### Step 7: Convert to Degrees
Since we need the angle in sexagesimal measure (degrees), we can use a calculator to find:
\[ θ \approx 0.0573 \text{ degrees} \]

### Step 8: Express in Terms of π
To express this in terms of π:
\[ θ \approx \frac{0.0573 \times 180}{\pi} \approx \frac{9}{50} \pi \text{ degrees} \]

### Conclusion
Thus, the angle subtended by the 1 m pole at a distance of 1 km on the ground is approximately:
\[ \theta \approx \frac{9}{50} \pi \text{ degrees} \]