Two towers of the same height stand on opposite sides of a road 100 m wide. At a point on the road between the towers. The elevations of the towers are `30^(@) and 45^(@)`. Find the height of the towers and the position of the point from the nearest tower :

Options

A. 36.6 m and 63.4 m
B. 63.3 m and 63.4 m
C. 66.3 m and 63.4 m
D. 36.6 m and 86. 4m

Answer

To solve the problem, we will use trigonometric ratios to find the height of the towers and the position of the point from the nearest tower.

### Step 1: Define the problem
Let the height of the towers be \( h \). Let the distance from the point on the road to the nearest tower (the tower with a \( 30^\circ \) elevation) be \( x \). Therefore, the distance to the other tower (with a \( 45^\circ \) elevation) will be \( 100 - x \).

### Step 2: Set up the equations using trigonometry
Using the tangent function, we can set up the following equations based on the angles of elevation:

1. For the tower with a \( 30^\circ \) elevation:
\[ \tan(30^\circ) = \frac{h}{x} \]
We know that \( \tan(30^\circ) = \frac{1}{\sqrt{3}} \), so:
\[ \frac{1}{\sqrt{3}} = \frac{h}{x} \quad \Rightarrow \quad h = \frac{x}{\sqrt{3}} \quad \text{(Equation 1)} \]

2. For the tower with a \( 45^\circ \) elevation:
\[ \tan(45^\circ) = \frac{h}{100 - x} \]
Since \( \tan(45^\circ) = 1 \), we have:
\[ 1 = \frac{h}{100 - x} \quad \Rightarrow \quad h = 100 - x \quad \text{(Equation 2)} \]

### Step 3: Equate the two expressions for height
From Equation 1 and Equation 2, we can set them equal to each other:
\[ \frac{x}{\sqrt{3}} = 100 - x \]

### Step 4: Solve for \( x \)
To solve for \( x \), first multiply both sides by \( \sqrt{3} \):
\[ x = \sqrt{3}(100 - x) \]
Expanding this gives:
\[ x = 100\sqrt{3} - \sqrt{3}x \]
Now, add \( \sqrt{3}x \) to both sides:
\[ x + \sqrt{3}x = 100\sqrt{3} \]
Factor out \( x \):
\[ x(1 + \sqrt{3}) = 100\sqrt{3} \]
Now, divide both sides by \( (1 + \sqrt{3}) \):
\[ x = \frac{100\sqrt{3}}{1 + \sqrt{3}} \]

### Step 5: Calculate \( h \)
Now that we have \( x \), we can substitute it back into either Equation 1 or Equation 2 to find \( h \). Using Equation 2:
\[ h = 100 - x = 100 - \frac{100\sqrt{3}}{1 + \sqrt{3}} \]
To simplify this, we can find a common denominator:
\[ h = \frac{(100 + 100\sqrt{3}) - 100\sqrt{3}}{1 + \sqrt{3}} = \frac{100}{1 + \sqrt{3}} \]

### Step 6: Final results
Now we have \( x \) and \( h \):
1. The height of the towers \( h = \frac{100}{1 + \sqrt{3}} \).
2. The distance from the nearest tower \( x = \frac{100\sqrt{3}}{1 + \sqrt{3}} \).