A statue of height 4 m stands on a tower of height 10 m. The angle subtended by the status at the eyes of an observer of height 2m, standing at a distance of 6m from base of the tower is

Options

A. `tan^(-1)((2)/(11))`
B. `tan^(-1)((4)/(3))`
C. `tan^(-1)(2)`
D. `tan^(-1)((7)/(13))`

Answer

To solve the problem step by step, we need to calculate the angle subtended by the statue at the observer's eye level. Here’s how we can approach it:

### Step 1: Understand the heights and distances
- Height of the tower = 10 m
- Height of the statue = 4 m
- Height of the observer = 2 m
- Distance of the observer from the base of the tower = 6 m

### Step 2: Calculate the total height of the statue above the observer's eye level
The total height from the ground to the top of the statue is:
\[ \text{Total height} = \text{Height of tower} + \text{Height of statue} = 10 \, \text{m} + 4 \, \text{m} = 14 \, \text{m} \]

The height of the observer's eyes is 2 m, so the height from the observer's eyes to the top of the statue is:
\[ \text{Height from observer's eye to top of statue} = \text{Total height} - \text{Height of observer} = 14 \, \text{m} - 2 \, \text{m} = 12 \, \text{m} \]

### Step 3: Set up the triangle
Now we have a right triangle where:
- The opposite side (height from observer's eye to top of statue) = 12 m
- The adjacent side (distance from the observer to the base of the tower) = 6 m

### Step 4: Calculate the angle using the tangent function
Using the tangent function:
\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{12}{6} = 2 \]

### Step 5: Find the angle
To find the angle \( \theta \):
\[ \theta = \tan^{-1}(2) \]

### Step 6: Calculate the angle subtended by the statue
Now we need to find the angle subtended by the statue at the observer's eye level. We will denote the angle subtended by the statue as \( \alpha \).

### Step 7: Calculate the angle subtended by the base of the statue
The height of the statue above the observer's eye level is:
\[ \text{Height of statue above observer's eye} = 4 \, \text{m} - 2 \, \text{m} = 2 \, \text{m} \]

Using the tangent function again for the angle \( \alpha \):
\[ \tan(\alpha) = \frac{\text{height of statue above observer's eye}}{\text{distance from observer to base of tower}} = \frac{2}{6} = \frac{1}{3} \]

### Step 8: Find the angle \( \alpha \)
To find the angle \( \alpha \):
\[ \alpha = \tan^{-1}\left(\frac{1}{3}\right) \]

### Final Answer
Thus, the angle subtended by the statue at the observer's eye level is:
\[ \alpha = \tan^{-1}\left(\frac{1}{3}\right) \]