The angle of elevation of the top of a pillar at any point A on the ground is `15^(@)`. On walking 100 ft. towards the pillar, the angle becomes `30^(@)`. Height of the pillar and its distance from A are ____and____respectively.

Answer

To solve the problem, we need to find the height of the pillar and the distance from point A to the base of the pillar. Let's break down the steps:

### Step 1: Understand the Problem
We have a pillar and two points, A and B. The angle of elevation from point A to the top of the pillar is 15 degrees. After walking 100 feet towards the pillar to point B, the angle of elevation becomes 30 degrees. We need to find the height of the pillar (H) and the distance from point A to the base of the pillar (D).

### Step 2: Draw the Diagram
1. Draw a vertical line representing the pillar (CD).
2. Mark point A on the ground at a distance D from the base of the pillar.
3. Mark point B, which is 100 feet closer to the pillar than point A.
4. Label the angles of elevation from A to the top of the pillar as 15 degrees and from B as 30 degrees.

### Step 3: Set Up the Equations
Using the tangent function for the angles of elevation:
1. From point B:
\[ \tan(30^\circ) = \frac{H}{D - 100} \]
Since \(\tan(30^\circ) = \frac{1}{\sqrt{3}}\), we can write:
\[ \frac{1}{\sqrt{3}} = \frac{H}{D - 100} \quad \text{(1)} \]

2. From point A:
\[ \tan(15^\circ) = \frac{H}{D} \]
Since \(\tan(15^\circ) = 2 - \sqrt{3}\), we can write:
\[ 2 - \sqrt{3} = \frac{H}{D} \quad \text{(2)} \]

### Step 4: Solve for H and D
From equation (1), we can express H in terms of D:
\[ H = (D - 100) \cdot \frac{1}{\sqrt{3}} \quad \text{(3)} \]

From equation (2), we can express H in terms of D:
\[ H = D(2 - \sqrt{3}) \quad \text{(4)} \]

### Step 5: Equate the Two Expressions for H
Set equations (3) and (4) equal to each other:
\[ (D - 100) \cdot \frac{1}{\sqrt{3}} = D(2 - \sqrt{3}) \]

### Step 6: Solve for D
Multiply both sides by \(\sqrt{3}\) to eliminate the fraction:
\[ D - 100 = D\sqrt{3}(2 - \sqrt{3}) \]
Rearranging gives:
\[ D - D\sqrt{3}(2 - \sqrt{3}) = 100 \]
Factoring out D:
\[ D(1 - \sqrt{3}(2 - \sqrt{3})) = 100 \]
Calculating \(\sqrt{3}(2 - \sqrt{3})\):
\[ \sqrt{3} \cdot 2 - 3 = 2\sqrt{3} - 3 \]
Thus:
\[ D(1 - (2\sqrt{3} - 3)) = 100 \]
\[ D(4 - 2\sqrt{3}) = 100 \]
Now solve for D:
\[ D = \frac{100}{4 - 2\sqrt{3}} \]

### Step 7: Rationalize the Denominator
Multiply the numerator and denominator by the conjugate:
\[ D = \frac{100(4 + 2\sqrt{3})}{(4 - 2\sqrt{3})(4 + 2\sqrt{3})} \]
Calculating the denominator:
\[ (4 - 2\sqrt{3})(4 + 2\sqrt{3}) = 16 - 12 = 4 \]
Thus:
\[ D = 25(4 + 2\sqrt{3}) = 100 + 50\sqrt{3} \]

### Step 8: Find H
Substituting D back into equation (4):
\[ H = D(2 - \sqrt{3}) = (100 + 50\sqrt{3})(2 - \sqrt{3}) \]
Calculating:
\[ H = 200 + 100\sqrt{3} - 100 - 50\sqrt{3} = 100 + 50\sqrt{3} \]

### Final Answers
- Height of the pillar (H) = \(100 + 50\sqrt{3}\) feet
- Distance from A to the base of the pillar (D) = \(100 + 50\sqrt{3}\) feet