From a point on the ground , 20m away from the bottom of a building , the angle of elevation of the top of the building is `60^(@)`. Find the distance of the point from the top of the building.

Options

A. 10m
B. `40sqrt(3)m`
C. `40m`
D. `(20)/(sqrt(3))m`

Answer

To solve the problem step by step, we will use trigonometric ratios.

### Step 1: Understand the Problem
We have a right triangle formed by the height of the building, the distance from the point on the ground to the base of the building, and the line of sight from the point to the top of the building.

### Step 2: Identify the Given Values
- Distance from the point on the ground to the base of the building (AB) = 20 m
- Angle of elevation (∠CAB) = 60°

### Step 3: Set Up the Right Triangle
Let:
- A be the point on the ground,
- B be the base of the building,
- C be the top of the building.

We need to find the distance AC, which is the hypotenuse of the triangle ABC.

### Step 4: Use Trigonometric Ratios
We can use the cosine of the angle to find the hypotenuse (AC):
\[ \cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}} \]
Here, the adjacent side is AB (20 m), and the hypotenuse is AC.

### Step 5: Substitute the Values
Substituting the known values into the cosine formula:
\[ \cos(60^\circ) = \frac{AB}{AC} \]
\[ \cos(60^\circ) = \frac{20}{AC} \]
We know that \(\cos(60^\circ) = \frac{1}{2}\).

### Step 6: Solve for AC
Now substituting \(\cos(60^\circ)\) into the equation:
\[ \frac{1}{2} = \frac{20}{AC} \]
Cross-multiplying gives:
\[ AC = 20 \times 2 = 40 \text{ m} \]

### Step 7: Conclusion
The distance from the point on the ground to the top of the building (AC) is 40 meters.