One files a kite with a thread 150 metre long .If the thread of the kite makes an angle of `60^(@)` with the horizontal line , then the height of the kite from the ground (as suming the thread to be in a straight line) is

Options

A. 50 metre
B. `75sqrt(3)` metre
C. `25sqrt(3)` metre
D. 80 metre

Answer

To solve the problem of finding the height of the kite from the ground, we can use the properties of a right triangle formed by the thread of the kite, the height of the kite, and the horizontal distance from the point where the thread is held to the point directly below the kite.

### Step-by-Step Solution:

1. **Identify the Triangle**:
- Let the point where the thread is held be point A, the point directly below the kite on the ground be point B, and the position of the kite be point C.
- We have a right triangle ABC, where:
- AC is the length of the thread (150 meters).
- Angle CAB is 60 degrees (the angle the thread makes with the horizontal).
- AB is the height of the kite from the ground (which we need to find).

2. **Use the Sine Function**:
- In a right triangle, the sine of an angle is defined as the ratio of the length of the opposite side to the hypotenuse.
- Here, we can use the sine function for angle CAB:
\[ \sin(60^\circ) = \frac{AB}{AC} \]
- We know that \( AC = 150 \) meters.

3. **Substitute the Values**:
- Substitute the known values into the sine function:
\[ \sin(60^\circ) = \frac{AB}{150} \]
- The value of \( \sin(60^\circ) \) is \( \frac{\sqrt{3}}{2} \).

4. **Set Up the Equation**:
- Now, we can set up the equation:
\[ \frac{\sqrt{3}}{2} = \frac{AB}{150} \]

5. **Solve for AB**:
- To find \( AB \), multiply both sides of the equation by 150:
\[ AB = 150 \cdot \frac{\sqrt{3}}{2} \]
- This simplifies to:
\[ AB = 75\sqrt{3} \]

6. **Calculate the Height**:
- Now, we can calculate the numerical value of \( 75\sqrt{3} \):
\[ AB \approx 75 \cdot 1.732 \approx 129.9 \text{ meters} \]

Thus, the height of the kite from the ground is approximately \( 75\sqrt{3} \) meters or about \( 129.9 \) meters.