80 m away from the foot of the tower, the angle of elevation of the top of the tower is `60^@`. What is the height (in metres) of the tower?

Options

A. 40
B. `60sqrt3`
C. `80sqrt3`
D. `40//sqrt3`

Answer

To find the height of the tower given the distance from the foot of the tower and the angle of elevation, we can use trigonometric ratios. Here’s a step-by-step solution:

### Step 1: Understand the Problem
We are given:
- Distance from the foot of the tower (BC) = 80 m
- Angle of elevation (∠ABC) = 60°

We need to find the height of the tower (AB).

### Step 2: Draw a Diagram
Draw a right triangle where:
- Point A is the top of the tower.
- Point B is the foot of the tower.
- Point C is the point on the ground 80 m away from B.

### Step 3: Identify the Right Triangle
In triangle ABC:
- AB is the height of the tower (perpendicular).
- BC is the distance from the foot of the tower (base).
- AC is the hypotenuse.

### Step 4: Use the Tangent Function
In a right triangle, the tangent of an angle is defined as the ratio of the opposite side (height of the tower) to the adjacent side (distance from the tower). Thus, we can write:

\[ \tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{AB}{BC} \]

Substituting the known values:

\[ \tan(60^\circ) = \frac{AB}{80} \]

### Step 5: Find the Value of Tangent
The value of \(\tan(60^\circ)\) is \(\sqrt{3}\). Therefore, we can substitute this value into the equation:

\[ \sqrt{3} = \frac{AB}{80} \]

### Step 6: Solve for AB
To find the height of the tower (AB), rearrange the equation:

\[ AB = 80 \cdot \sqrt{3} \]

### Step 7: Calculate the Height
Now, we can calculate the height:

\[ AB = 80 \cdot \sqrt{3} \approx 138.56 \text{ m} \]

However, since the question asks for the height in meters, we can leave it as \(80\sqrt{3}\) m.

### Final Answer
The height of the tower is \(80\sqrt{3}\) meters.

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