The angular depression of the top and the foot of the chimney seen from the top of a tower on the same base level as the chimney are `tan^-1(4/3)` and `tan^-1(5/2)` respectively if the height of the tower is 150m. then the distance between the top of the chimney and the tower is

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the Geometry
We have a tower (AB) and a chimney (CD) on the same base level. The height of the tower (AB) is given as 150 m. We need to find the distance between the top of the chimney (C) and the tower (A).

### Step 2: Set Up the Angles
The angles of depression from the top of the tower to the top and foot of the chimney are given as:
- Angle of depression to the top of the chimney (C): \( \alpha = \tan^{-1}\left(\frac{4}{3}\right) \)
- Angle of depression to the foot of the chimney (D): \( \beta = \tan^{-1}\left(\frac{5}{2}\right) \)

### Step 3: Find the Distance to the Foot of the Chimney
Using the angle \( \beta \):
\[ \tan(\beta) = \frac{\text{height of tower}}{\text{distance to foot of chimney}} \]
Let the distance from the tower to the foot of the chimney (D) be \( y \):
\[ \tan\left(\tan^{-1}\left(\frac{5}{2}\right)\right) = \frac{150}{y} \]
This gives:
\[ \frac{5}{2} = \frac{150}{y} \]
Cross-multiplying gives:
\[ 5y = 300 \implies y = \frac{300}{5} = 60 \text{ m} \]

### Step 4: Find the Distance to the Top of the Chimney
Using the angle \( \alpha \):
\[ \tan(\alpha) = \frac{\text{height of tower}}{\text{distance to top of chimney}} \]
Let the distance from the tower to the top of the chimney (C) be \( x \):
\[ \tan\left(\tan^{-1}\left(\frac{4}{3}\right)\right) = \frac{150}{x} \]
This gives:
\[ \frac{4}{3} = \frac{150}{x} \]
Cross-multiplying gives:
\[ 4x = 450 \implies x = \frac{450}{4} = 112.5 \text{ m} \]

### Step 5: Calculate the Distance Between the Top of the Chimney and the Tower
Now we can find the distance \( AE \) (the distance from the top of the chimney to the tower) using the Pythagorean theorem:
\[ AE^2 = x^2 + y^2 \]
Substituting the values:
\[ AE^2 = (112.5)^2 + (60)^2 \]
Calculating:
\[ AE^2 = 12656.25 + 3600 = 16256.25 \]
Taking the square root:
\[ AE = \sqrt{16256.25} \approx 127.5 \text{ m} \]

### Final Answer
The distance between the top of the chimney and the tower is approximately \( 127.5 \text{ m} \).

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