Two points A and B are on the ground and on opposite sides of a tower. A is closer to the foot of tower by 42 m than B. If the angles of elevation ofthe top of the tower, as observed from A and B are 60∘ and 45∘, respectively. then the height of the tower is closest to:

Options

A. 87.6 m
B. 98.6 m
C. 88.2 m
D. 99.4 m

Answer

To solve the problem step by step, we will use trigonometry and the information given about the angles of elevation from points A and B.

### Step 1: Define Variables
Let:
- \( h \) = height of the tower
- \( x \) = distance from point B to the foot of the tower
- \( x - 42 \) = distance from point A to the foot of the tower (since A is 42 m closer than B)

### Step 2: Set Up the Equations
From point B, the angle of elevation to the top of the tower is \( 45^\circ \):
Using the tangent function:
\[ \tan(45^\circ) = \frac{h}{x} \]
Since \( \tan(45^\circ) = 1 \):
\[ h = x \quad \text{(Equation 1)} \]

From point A, the angle of elevation to the top of the tower is \( 60^\circ \):
Using the tangent function again:
\[ \tan(60^\circ) = \frac{h}{x - 42} \]
Since \( \tan(60^\circ) = \sqrt{3} \):
\[ h = \sqrt{3} (x - 42) \quad \text{(Equation 2)} \]

### Step 3: Substitute Equation 1 into Equation 2
Substituting \( h = x \) from Equation 1 into Equation 2:
\[ x = \sqrt{3} (x - 42) \]

### Step 4: Solve for \( x \)
Expanding the equation:
\[ x = \sqrt{3}x - 42\sqrt{3} \]
Rearranging gives:
\[ x - \sqrt{3}x = -42\sqrt{3} \]
Factoring out \( x \):
\[ x(1 - \sqrt{3}) = -42\sqrt{3} \]
Now, solving for \( x \):
\[ x = \frac{-42\sqrt{3}}{1 - \sqrt{3}} \]

### Step 5: Rationalize the Denominator
To simplify \( x \):
Multiply the numerator and denominator by the conjugate of the denominator:
\[ x = \frac{-42\sqrt{3}(1 + \sqrt{3})}{(1 - \sqrt{3})(1 + \sqrt{3})} \]
Calculating the denominator:
\[ (1 - \sqrt{3})(1 + \sqrt{3}) = 1 - 3 = -2 \]
Thus:
\[ x = \frac{-42\sqrt{3}(1 + \sqrt{3})}{-2} = 21\sqrt{3}(1 + \sqrt{3}) \]

### Step 6: Calculate \( h \)
Now substitute \( x \) back into Equation 1 to find \( h \):
\[ h = x = 21\sqrt{3}(1 + \sqrt{3}) \]
Calculating \( 1 + \sqrt{3} \approx 1 + 1.732 = 2.732 \):
\[ h \approx 21 \cdot 1.732 \cdot 2.732 \]
Calculating \( 21 \cdot 1.732 \approx 36.372 \):
\[ h \approx 36.372 \cdot 2.732 \approx 99.5 \text{ m} \]

### Step 7: Round to the Closest Integer
The height of the tower is approximately \( 99.5 \) m, which rounds to \( 100 \) m.

### Final Answer
The height of the tower is closest to **100 m**.