A flag leaning to wards the east is inclined at an angle `theta` to the ground. A man walks a distance of 4 m from the foot of the flag towards the west and the observes the angle of elevation of the top of the flag to be `45^(@)`. If, on walking a further distance of 3 m in the same direction, the angle of elevation decreases by `15^(@), tan theta` is equal to

Options

A. `1(3+4sqrt(3))`
B. `3(7+4sqrt(3))`
C. `4(7+3sqrt(3))`
D. none of these

Answer

To solve the problem step by step, we will use the information given about the flag, the angles of elevation, and the distances walked by the man.

### Step 1: Define the Variables
Let:
- \( h \) = height of the flag
- \( d \) = distance from the foot of the flag to the man when he observes the angle of elevation of \( 45^\circ \)
- \( \theta \) = angle of inclination of the flag with the ground

From the problem, when the man is at the foot of the flag, he observes the angle of elevation to be \( 45^\circ \). This means that the height of the flag \( h \) is equal to the distance \( d \) from the man to the foot of the flag at that moment.

### Step 2: Set Up the First Equation
When the man walks 4 m towards the west, the distance from the foot of the flag becomes \( d + 4 \). At this position, the angle of elevation is \( 45^\circ \). Thus, we can write:
\[ h = d + 4 \]
Since \( \tan(45^\circ) = 1 \), we have:
\[ h = d + 4 \]

### Step 3: Set Up the Second Equation
After walking another 3 m (making it a total of 7 m from the foot of the flag), the angle of elevation decreases to \( 30^\circ \) (since it decreases by \( 15^\circ \)). Thus, we can write:
\[ \tan(30^\circ) = \frac{h}{d + 7} \]
Since \( \tan(30^\circ) = \frac{1}{\sqrt{3}} \), we have:
\[ \frac{h}{d + 7} = \frac{1}{\sqrt{3}} \]
This leads to:
\[ h = \frac{(d + 7)}{\sqrt{3}} \]

### Step 4: Equate the Two Expressions for \( h \)
Now we have two expressions for \( h \):
1. \( h = d + 4 \)
2. \( h = \frac{(d + 7)}{\sqrt{3}} \)

Setting these equal gives:
\[ d + 4 = \frac{(d + 7)}{\sqrt{3}} \]

### Step 5: Solve for \( d \)
To eliminate the fraction, multiply both sides by \( \sqrt{3} \):
\[ \sqrt{3}(d + 4) = d + 7 \]
Expanding this gives:
\[ \sqrt{3}d + 4\sqrt{3} = d + 7 \]
Rearranging terms:
\[ \sqrt{3}d - d = 7 - 4\sqrt{3} \]
Factoring out \( d \):
\[ d(\sqrt{3} - 1) = 7 - 4\sqrt{3} \]
Thus,
\[ d = \frac{7 - 4\sqrt{3}}{\sqrt{3} - 1} \]

### Step 6: Find \( h \)
Substituting \( d \) back into the first equation for \( h \):
\[ h = d + 4 = \frac{7 - 4\sqrt{3}}{\sqrt{3} - 1} + 4 \]

### Step 7: Calculate \( \tan(\theta) \)
Now, we need to find \( \tan(\theta) \). Since the flag is inclined at an angle \( \theta \) with the ground, we can use the height \( h \) and the distance \( d \):
\[ \tan(\theta) = \frac{h}{d} \]

### Step 8: Final Calculation
Substituting the values of \( h \) and \( d \) into the equation for \( \tan(\theta) \) will yield the final answer.