From 40 m away from the foot of a tower , the angle of elevation of the top of the tower is `60^(@)` .What is the height of the tower ?

Options

A. `(120)/(sqrt(3))` m .
B. `(60)/(sqrt(3))` m .
C. `(50)/(sqrt(3))` m .
D. `(130)/(sqrt(7))`m .

Answer

To find the height of the tower based on the given information, we can use trigonometric ratios. Here's a step-by-step solution:

### Step 1: Understand the Problem
We have a tower, and we are standing 40 meters away from its base. The angle of elevation to the top of the tower is 60 degrees. We need to find the height of the tower.

### Step 2: Identify the Right Triangle
We can visualize the situation as a right triangle where:
- The height of the tower is the opposite side (perpendicular).
- The distance from the foot of the tower to the point where we are standing is the adjacent side (base).
- The angle of elevation is 60 degrees.

### Step 3: Use the Tangent Function
The tangent of an angle in a right triangle is defined as the ratio of the opposite side to the adjacent side. Therefore, we can write:
\[ \tan(60^\circ) = \frac{\text{Height of the tower}}{\text{Distance from the tower}} \]
Substituting the known values:
\[ \tan(60^\circ) = \frac{h}{40} \]

### Step 4: Find the Value of \(\tan(60^\circ)\)
From trigonometric tables or the unit circle, we know:
\[ \tan(60^\circ) = \sqrt{3} \]

### Step 5: Substitute and Solve for Height
Now we can substitute the value of \(\tan(60^\circ)\) into the equation:
\[ \sqrt{3} = \frac{h}{40} \]
To find \(h\), multiply both sides by 40:
\[ h = 40 \cdot \sqrt{3} \]

### Step 6: Final Answer
Thus, the height of the tower is:
\[ h = 40\sqrt{3} \text{ meters} \]