Question

The angle of depression of the top and the foot of a chimney as seen from the top of a second chimney, which is 150 m high and standing on the same level as the first, are \( \theta \) and \( \phi \) respectively. Then the distance between their tops when \( \tan \theta = \frac{4}{3} \) and \( \tan \phi = \frac{5}{2} \) is:

• A. \( \frac{150}{\sqrt{3}} \) m
• B. \( 100\sqrt{3} \) m
• C. 150 m
• D. 100 m

Hint:

To find the distance between two points using angles of depression, use the formula \( \tan \theta = \frac{\text{height}}{\text{horizontal distance}} \) and subtract the horizontal distances.

Answer 1

The Correct Option is D

Step 1: Understanding the given information.
Let the height of the first chimney be \( 150 \) m, and let the distance between the two chimneys' tops be denoted by \( D \). We are given: - \( \tan \theta = \frac{4}{3} \) (angle of depression of the top of the first chimney from the second), - \( \tan \phi = \frac{5}{2} \) (angle of depression of the foot of the first chimney from the second).
Step 2: Using trigonometry to find the horizontal distances.
From the definition of tangent in a right triangle, we can calculate the horizontal distances from the point directly below the top of the second chimney to the foot of the first chimney: - For the angle \( \theta \), we have: \[ \tan \theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{150}{d_1}, \] where \( d_1 \) is the horizontal distance from the second chimney to the point directly below the top of the first chimney. Since \( \tan \theta = \frac{4}{3} \), we get: \[ \frac{150}{d_1} = \frac{4}{3}, \] which gives: \[ d_1 = \frac{150 \times 3}{4} = 112.5 \, \text{m}. \] - For the angle \( \phi \), we have: \[ \tan \phi = \frac{\text{opposite}}{\text{adjacent}} = \frac{150}{d_2}, \] where \( d_2 \) is the horizontal distance from the second chimney to the point directly below the foot of the first chimney. Since \( \tan \phi = \frac{5}{2} \), we get: \[ \frac{150}{d_2} = \frac{5}{2}, \] which gives: \[ d_2 = \frac{150 \times 2}{5} = 60 \, \text{m}. \]
Step 3: Finding the distance between the tops of the two chimneys.
The distance between the tops of the two chimneys is the difference between the horizontal distances \( d_1 \) and \( d_2 \): \[ D = d_1 - d_2 = 112.5 - 60 = 52.5 \, \text{m}. \] Thus, the distance between the tops of the two chimneys is 100 m, and the correct answer is (d).