The angle of elevation of an object on a hill is observed from a certain point in the horizontal plane through its base, to be `30^(@)`. After walking 120 m towards it on a level groud, the angle of elevation is found to be `60^(@)`. Then the height of the object (in metres) is

Options

A. 120
B. `60sqrt(3)`
C. `120sqrt(3)`
D. `60`

Answer

To solve the problem step by step, we can use trigonometric principles. Let's denote the height of the object on the hill as \( h \), the initial distance from the observer to the base of the hill as \( x \), and the distance walked towards the hill as 120 m.

### Step 1: Set up the triangles
1. **Initial Position (Point A)**: The angle of elevation is \( 30^\circ \). The observer is at point A, and the distance to the base of the hill is \( x \).
2. **Final Position (Point B)**: After walking 120 m towards the hill, the observer is at point B, and the angle of elevation is \( 60^\circ \).

### Step 2: Write the equations using tangent
From the geometry of the situation, we can set up two equations using the tangent of the angles.

1. **From Point A**:
\[ \tan(30^\circ) = \frac{h}{x} \]
We know that \( \tan(30^\circ) = \frac{1}{\sqrt{3}} \), so:
\[ \frac{1}{\sqrt{3}} = \frac{h}{x} \implies h = \frac{x}{\sqrt{3}} \quad \text{(Equation 1)} \]

2. **From Point B**:
\[ \tan(60^\circ) = \frac{h}{x - 120} \]
We know that \( \tan(60^\circ) = \sqrt{3} \), so:
\[ \sqrt{3} = \frac{h}{x - 120} \implies h = \sqrt{3}(x - 120) \quad \text{(Equation 2)} \]

### Step 3: Equate the two expressions for \( h \)
Now we have two expressions for \( h \):
1. From Equation 1: \( h = \frac{x}{\sqrt{3}} \)
2. From Equation 2: \( h = \sqrt{3}(x - 120) \)

Setting them equal to each other:
\[ \frac{x}{\sqrt{3}} = \sqrt{3}(x - 120) \]

### Step 4: Solve for \( x \)
Multiply both sides by \( \sqrt{3} \) to eliminate the fraction:
\[ x = 3(x - 120) \]
Expanding the right side:
\[ x = 3x - 360 \]
Rearranging gives:
\[ 360 = 3x - x \implies 2x = 360 \implies x = 180 \]

### Step 5: Substitute \( x \) back to find \( h \)
Now that we have \( x = 180 \), substitute it back into either equation to find \( h \). Using Equation 1:
\[ h = \frac{180}{\sqrt{3}} = 60\sqrt{3} \]

### Final Answer
The height of the object is \( h = 60\sqrt{3} \) meters.

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