A flagstaff on the top of tower 80 m high, subtends an angle `tan^(-1)(1/9)` at point on the ground 100 m from the tower. The height of flagstaff is

Answer

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

**Step 1: Understand the problem and draw a diagram.**
- We have a tower of height 80 m with a flagstaff on top.
- A point on the ground is 100 m away from the base of the tower.
- The angle subtended by the flagstaff at this point is given as \( \tan^{-1}\left(\frac{1}{9}\right) \).

**Step 2: Define the variables.**
- Let \( AB \) be the height of the flagstaff.
- The total height from the ground to the top of the flagstaff is \( AC = AB + 80 \).

**Step 3: Use the tangent function for the angle at the point on the ground.**
- From the point on the ground, the angle \( \alpha = \tan^{-1}\left(\frac{1}{9}\right) \).
- Therefore, \( \tan(\alpha) = \frac{1}{9} \).

**Step 4: Set up the triangle formed by the tower and the point on the ground.**
- In triangle \( ACD \) (where \( D \) is the point on the ground):
\[ \tan(\alpha) = \frac{AC}{CD} \]
Here, \( CD = 100 \) m (distance from the tower) and \( AC = AB + 80 \).

**Step 5: Substitute the known values into the tangent equation.**
- We can write:
\[ \frac{AC}{100} = \frac{1}{9} \]
Therefore, \( AC = \frac{100}{9} \).

**Step 6: Relate \( AC \) to \( AB \).**
- Since \( AC = AB + 80 \), we substitute:
\[ AB + 80 = \frac{100}{9} \]
- Rearranging gives:
\[ AB = \frac{100}{9} - 80 \]

**Step 7: Calculate \( AB \).**
- Convert 80 to a fraction with a common denominator:
\[ 80 = \frac{720}{9} \]
- Now, substitute:
\[ AB = \frac{100}{9} - \frac{720}{9} = \frac{100 - 720}{9} = \frac{-620}{9} \]
- This result is incorrect since height cannot be negative. Let's check the calculations again.

**Step 8: Correct the calculation.**
- We need to find \( \tan(\beta) \) for the triangle \( BCD \):
- The height of the tower is 80 m, and the distance from the tower is 100 m.
- Therefore, \( \tan(\beta) = \frac{80}{100} = \frac{4}{5} \).

**Step 9: Use the tangent addition formula.**
- We know:
\[ \tan(\alpha + \beta) = \frac{\tan(\alpha) + \tan(\beta)}{1 - \tan(\alpha)\tan(\beta)} \]
- Substitute the values:
\[ \tan(\alpha + \beta) = \frac{\frac{1}{9} + \frac{4}{5}}{1 - \left(\frac{1}{9} \cdot \frac{4}{5}\right)} \]

**Step 10: Calculate \( \tan(\alpha + \beta) \).**
- Find a common denominator for the numerator:
\[ \frac{1}{9} + \frac{4}{5} = \frac{5 + 36}{45} = \frac{41}{45} \]
- Calculate the denominator:
\[ 1 - \frac{4}{45} = \frac{41}{45} \]
- Thus,
\[ \tan(\alpha + \beta) = \frac{\frac{41}{45}}{\frac{41}{45}} = 1 \]

**Step 11: Relate back to the height.**
- From the triangle \( ACD \):
\[ \tan(\alpha + \beta) = \frac{AB + 80}{100} \]
- Setting this equal to 1 gives:
\[ AB + 80 = 100 \implies AB = 100 - 80 = 20 \text{ m} \]

**Final Answer:**
The height of the flagstaff is \( 20 \) m.

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