The height f a chimney when it is found that on walking towards it 100 ft. in a horizontal line through its base the angular elevation of its top changes from `30^(@)` to `45^(@)` is____

Answer

To find the height of the chimney, we can break down the problem step by step.

### Step 1: Understand the Problem
We have a chimney and two points from which we observe the top of the chimney. The angles of elevation from these two points are given as 30 degrees and 45 degrees. The distance moved towards the chimney is 100 ft.

### Step 2: Set Up the Diagram
Let's denote:
- The height of the chimney as \( H \).
- The distance from the base of the chimney to the point where the angle of elevation is 30 degrees as \( x \).
- The distance from the base of the chimney to the point where the angle of elevation is 45 degrees will then be \( x - 100 \).

### Step 3: Use Trigonometric Ratios
For the first observation point (where the angle is 30 degrees):
\[ \tan(30^\circ) = \frac{H}{x} \]
We know that \( \tan(30^\circ) = \frac{1}{\sqrt{3}} \), so:
\[ \frac{1}{\sqrt{3}} = \frac{H}{x} \implies H = \frac{x}{\sqrt{3}} \tag{1} \]

For the second observation point (where the angle is 45 degrees):
\[ \tan(45^\circ) = \frac{H}{x - 100} \]
Since \( \tan(45^\circ) = 1 \), we have:
\[ 1 = \frac{H}{x - 100} \implies H = x - 100 \tag{2} \]

### Step 4: Solve the Equations
Now we have two equations:
1. \( H = \frac{x}{\sqrt{3}} \)
2. \( H = x - 100 \)

Setting the two expressions for \( H \) equal to each other:
\[ \frac{x}{\sqrt{3}} = x - 100 \]

### Step 5: Rearranging the Equation
Multiply both sides by \( \sqrt{3} \) to eliminate the fraction:
\[ x = \sqrt{3}(x - 100) \]
Expanding the right side:
\[ x = \sqrt{3}x - 100\sqrt{3} \]
Rearranging gives:
\[ x - \sqrt{3}x = -100\sqrt{3} \]
Factoring out \( x \):
\[ x(1 - \sqrt{3}) = -100\sqrt{3} \]
Thus,
\[ x = \frac{-100\sqrt{3}}{1 - \sqrt{3}} \tag{3} \]

### Step 6: Substitute Back to Find \( H \)
Now substitute \( x \) from equation (3) into either equation (1) or (2) to find \( H \). Using equation (2):
\[ H = x - 100 \]
Substituting for \( x \):
\[ H = \frac{-100\sqrt{3}}{1 - \sqrt{3}} - 100 \]

### Step 7: Simplifying \( H \)
To simplify, we can find a common denominator:
\[ H = \frac{-100\sqrt{3} - 100(1 - \sqrt{3})}{1 - \sqrt{3}} = \frac{-100\sqrt{3} - 100 + 100\sqrt{3}}{1 - \sqrt{3}} = \frac{-100}{1 - \sqrt{3}} \]
Now rationalizing the denominator:
\[ H = \frac{-100(1 + \sqrt{3})}{(1 - \sqrt{3})(1 + \sqrt{3})} = \frac{-100(1 + \sqrt{3})}{1 - 3} = \frac{-100(1 + \sqrt{3})}{-2} = 50(1 + \sqrt{3}) \]

### Final Answer
Thus, the height of the chimney is:
\[ H = 50(1 + \sqrt{3}) \text{ feet} \]