The angular depressions of the top and the foot of a chimney as seen from the top of a second chimney which is 150 meters high and standing on the same level as the first are `theta` and `phi` respectively. Find the distance between their tops, when `tan theta = 4/3, tan phi = 5/2`.

Answer

To solve the problem, we need to find the distance between the tops of two chimneys based on the angular depressions given. Let's break it down step by step.

### Step 1: Understanding the Problem
We have two chimneys. The height of the second chimney (from which we are observing) is 150 meters. The angles of depression to the top and bottom of the first chimney are given as θ and φ respectively, where:
- tan(θ) = 4/3
- tan(φ) = 5/2

### Step 2: Setting Up the Diagram
Let’s denote:
- A = Top of the second chimney (150 m high)
- B = Bottom of the first chimney
- C = Top of the first chimney
- D = Foot of the first chimney

From point A, we have:
- The angle of depression to point C (top of the first chimney) is θ.
- The angle of depression to point B (bottom of the first chimney) is φ.

### Step 3: Using the Tangent Function
Using the tangent function for the angles of depression, we can set up two equations based on the right triangles formed.

1. For angle φ (to the bottom of the first chimney):
\[ \tan(φ) = \frac{\text{Height of second chimney}}{\text{Horizontal distance (y)}} \]
\[ \tan(φ) = \frac{150}{y} \]
Given that \(\tan(φ) = \frac{5}{2}\):
\[ \frac{5}{2} = \frac{150}{y} \]
Rearranging gives:
\[ y = \frac{150 \times 2}{5} = 60 \text{ meters} \]

2. For angle θ (to the top of the first chimney):
\[ \tan(θ) = \frac{\text{Height of second chimney - Height of first chimney}}{\text{Horizontal distance (x)}} \]
\[ \tan(θ) = \frac{150 - h}{x} \]
Given that \(\tan(θ) = \frac{4}{3}\):
\[ \frac{4}{3} = \frac{150 - h}{x} \]
Rearranging gives:
\[ 4x = 3(150 - h) \]
We will find h later.

### Step 4: Finding the Height of the First Chimney
From the previous step, we need to express h in terms of x. We can also use the relationship between x and y:
\[ x = \frac{3}{4}y \]
Substituting y = 60:
\[ x = \frac{3}{4} \times 60 = 45 \text{ meters} \]

### Step 5: Finding the Height of the First Chimney
Now, we can substitute x back into the equation for θ:
\[ 4(45) = 3(150 - h) \]
\[ 180 = 450 - 3h \]
\[ 3h = 450 - 180 = 270 \]
\[ h = \frac{270}{3} = 90 \text{ meters} \]

### Step 6: Finding the Distance Between the Tops
Now we have:
- Height of the first chimney (h) = 90 m
- Height of the second chimney = 150 m
- Horizontal distances: x = 45 m, y = 60 m

The distance (d) between the tops of the two chimneys can be calculated using the Pythagorean theorem:
\[ d = \sqrt{(x^2 + (150 - 90)^2)} \]
\[ d = \sqrt{(45^2 + 60^2)} \]
Calculating:
\[ d = \sqrt{2025 + 3600} = \sqrt{5625} = 75 \text{ meters} \]

### Final Answer
The distance between the tops of the two chimneys is **75 meters**.