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 step by step, let's break it down:

### Step 1: Understand the Problem
We have two chimneys. The height of the second chimney (from which we are observing) is 150 meters. We need to find the distance between the tops of the two chimneys given the angles of depression (θ for the top of the first chimney and φ for the foot of the first chimney).

### Step 2: Draw the Diagram
1. Draw two vertical lines representing the two chimneys.
2. Label the height of the second chimney as 150 meters.
3. Label the top of the first chimney as point A and the foot of the first chimney as point B.
4. Draw lines from the top of the second chimney (point C) to point A and point B, forming angles θ and φ with the horizontal.

### Step 3: Identify the Variables
- Let the height of the first chimney be \( h \).
- The distance from the base of the second chimney to the base of the first chimney is \( d \).
- The height from the top of the second chimney to the top of the first chimney is \( 150 - h \).

### Step 4: Use the Tangent Function
From the angle of depression, we can use the tangent function:
1. For angle φ (to the foot of the first chimney):
\[ \tan(\phi) = \frac{150}{d} \]
Given \( \tan(\phi) = \frac{5}{2} \):
\[ \frac{5}{2} = \frac{150}{d} \]
Cross-multiplying gives:
\[ d = \frac{150 \cdot 2}{5} = 60 \text{ meters} \]

2. For angle θ (to the top of the first chimney):
\[ \tan(\theta) = \frac{150 - h}{d} \]
Given \( \tan(\theta) = \frac{4}{3} \):
\[ \frac{4}{3} = \frac{150 - h}{60} \]
Cross-multiplying gives:
\[ 4 \cdot 60 = 3(150 - h) \]
\[ 240 = 450 - 3h \]
Rearranging gives:
\[ 3h = 450 - 240 = 210 \]
\[ h = \frac{210}{3} = 70 \text{ meters} \]

### Step 5: Find the Height of the First Chimney
The height of the first chimney is \( h = 70 \) meters.

### Step 6: Calculate the Distance Between the Tops
To find the distance between the tops of the two chimneys (AE):
1. The height from the top of the first chimney to the top of the second chimney is:
\[ 150 - h = 150 - 70 = 80 \text{ meters} \]
2. Using the Pythagorean theorem in triangle ACD:
\[ AE^2 = (150 - h)^2 + d^2 \]
\[ AE^2 = 80^2 + 60^2 \]
\[ AE^2 = 6400 + 3600 = 10000 \]
\[ AE = \sqrt{10000} = 100 \text{ meters} \]

### Final Answer
The distance between the tops of the two chimneys is **100 meters**.

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