Question

The string of a flying kite is tied to a point on the ground. The length of the string between the kite and the point on the ground is 80 m. The string makes an angle of $30^\circ$ with the ground. The height of the kite above the ground is :

• A. $20\sqrt{3}$ m
• B. 40 m
• C. $40\sqrt{3}$ m
• D. $80\sqrt{3}$ m

Hint:

In a $30^\circ - 60^\circ - 90^\circ$ right triangle:
The side opposite to the $30^\circ$ angle is always exactly half the length of the hypotenuse!
Since the hypotenuse is 80 m, the height must be $80 / 2 = 40 \text{ m}$. This can be answered in seconds without written calculations!

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
This is a word problem on heights and distances.
We need to find the vertical height of a kite above the ground.
We are given:
- The length of the string (hypotenuse of the right-angled triangle) $= 80 \text{ m}$.
- The angle of elevation of the string with the ground $= 30^\circ$.

Step 2: Key Formula or Approach:
Let us model the situation as a right-angled triangle $\Delta \text{ABC}$:
- Let $C$ be the point on the ground where the string is tied.
- Let $A$ be the position of the kite in the air.
- Let $B$ be the point on the ground directly beneath the kite.
- Thus, $\angle\text{ABC} = 90^\circ$, and the angle of elevation is $\angle\text{ACB} = 30^\circ$.
- The length of the string is the hypotenuse, $\text{AC} = 80 \text{ m}$.
- The height of the kite is the opposite side, $\text{AB} = h$.
The trigonometric ratio that relates the opposite side and the hypotenuse is the sine function:
\[ \sin\theta = \frac{\text{Opposite}}{\text{Hypotenuse}} \]

Step 3: Detailed Explanation:

• Set up the sine trigonometric ratio for the right triangle:
\[ \sin(\angle\text{ACB}) = \frac{\text{AB}}{\text{AC}} \]

• Substitute the given values ($\theta = 30^\circ$, $\text{AC} = 80$, and $\text{AB} = h$):
\[ \sin 30^\circ = \frac{h}{80} \]

• Use the standard trigonometric value $\sin 30^\circ = \frac{1}{2}$:
\[ \frac{1}{2} = \frac{h}{80} \]

• Solve for the height $h$ by multiplying both sides by 80:
\[ h = 80 \times \frac{1}{2} \]
\[ h = 40 \text{ m} \]

• Thus, the vertical height of the kite above the ground is exactly 40 meters.


Step 4: Final Answer:
The height of the kite above the ground is 40 m.
Hence, option (B) is correct.