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 will first visualize the scenario and then calculate the required angle subtended by the statue at the observer's eyes.

### Step 1: Understand the Geometry
- The height of the tower (BC) is 10 m.
- The height of the statue (AB) is 4 m.
- Therefore, the total height from the ground to the top of the statue (A) is:
\[ AC = AB + BC = 4\,m + 10\,m = 14\,m \]
- The observer (D) is standing at a distance of 6 m from the base of the tower (point C) and is 2 m tall (height DE).

### Step 2: Calculate the Effective Height of the Statue from the Observer's Eyes
- The height of the observer's eyes (E) is 2 m. Thus, the height from the observer's eyes to the top of the statue (A) is:
\[ AF = AC - DE = 14\,m - 2\,m = 12\,m \]

### Step 3: Calculate the Angle Subtended by the Statue at the Observer's Eyes
- We need to find the angles \(\alpha\) (angle subtended by the top of the statue) and \(\beta\) (angle subtended by the bottom of the statue).
- The height from the observer's eyes to the base of the statue (B) is:
\[ BF = BC - DE = 10\,m - 2\,m = 8\,m \]

### Step 4: Use Trigonometry to Find \(\tan \alpha\) and \(\tan \beta\)
- For angle \(\alpha\):
\[ \tan \alpha = \frac{AF}{CD} = \frac{12\,m}{6\,m} = 2 \]

- For angle \(\beta\):
\[ \tan \beta = \frac{BF}{CD} = \frac{8\,m}{6\,m} = \frac{4}{3} \]

### Step 5: Calculate the Angle \(\alpha - \beta\)
- We can use the formula for the tangent of the difference of two angles:
\[ \tan(\alpha - \beta) = \frac{\tan \alpha - \tan \beta}{1 + \tan \alpha \cdot \tan \beta} \]

- Substituting the values:
\[ \tan(\alpha - \beta) = \frac{2 - \frac{4}{3}}{1 + 2 \cdot \frac{4}{3}} = \frac{\frac{6}{3} - \frac{4}{3}}{1 + \frac{8}{3}} = \frac{\frac{2}{3}}{\frac{11}{3}} = \frac{2}{11} \]

### Step 6: Find the Angle
- Therefore, the angle subtended by the statue at the observer's eyes is:
\[ \alpha - \beta = \tan^{-1}\left(\frac{2}{11}\right) \]

### Final Answer
The angle subtended by the statue at the eyes of the observer is \(\tan^{-1}\left(\frac{2}{11}\right)\).

---